POSNER, Chief Judge.

We write to untie a procedural knot partly of our own tying. Jays Food and a local of the chemical workers union have a collective bargaining agreement that provides for arbitration of disputes arising under it. In 1997 an arbitrator determined that Jays had violated the agreement by starting a third shift in a manner that reduced the paid hours of work of employees on the first and second shifts. Jays filed suit in federal district court to vacate the arbitrator's decision, and the union counterclaimed to confirm it. 9 U.S.C. § 9; 29 U.S.C. § 185. The district court ruled that the arbitrator had exceeded his authority under the collective bargaining agreement, and remanded the matter to the arbitrator "for further determination," adding that by doing this the court was "terminating case." After the arbitrator rendered his decision on remand, this time in favor of Jays, the union appealed from the order of remand. We dismissed the appeal in an unpublished order, on the ground that an order by a district court remanding a case to an arbitrator is not a final, appealable decision within the meaning of 28 U.S.C. § 1291 unless the remand requires a merely ministerial act, such as calculating interest when the period for which the calculation must be made, the interest rate, the principal to which the rate is to be applied, and whether interest is simple or compound have all been determined. Shearson Loeb Rhoades, Inc. v. Much, 754 F.2d 773, 776-78 (7th Cir. 1985); United Steelworkers v. Aurora Equipment Co., 830 F.2d 753, 754 (7th Cir. 1987). We had pointed out in these cases that if more is required, allowing an appeal would delay the ultimate resolution of the parties' dispute. More was required here. The arbitrator had written an opinion that did not satisfy the district judge, but the judge sent it back to him to make a new award rather than telling him to rule in favor of Jays.

It is true that by saying "terminating case" the district court might have been thought to be issuing a final judgment; but all he might have meant was that he was terminating the immediate proceeding before him, that is, the motion to vacate the arbitrator's award and the countermotion to confirm it. And whatever he meant, that may be all he did. For as we said in Chrysler Motors Corp. v. International Union, Allied Industrial Workers of America, 909 F.2d 248, 249 (7th Cir. 1990) (emphasis added), "The ground of the motion to dismiss the employer's appeal is that the order confirming the arbitrator's award is not final, since the matter of back pay remains to be determined by him. It is final in the district court, but under our cases that may not be good enough. Provided that the matter left for determination is not merely ministerial, . . . an order that does not determine the entire substantive relief to which the plaintiff is entitled is not a final decision even if the loose ends remain to be tied up by a decision-maker other than the district judge."

But we were wrong, not in general but with reference to the appealability of orders remanding cases to arbitrators, in thinking that the district court's order of remand was not immediately appealable. Congress had in 1988 amended the Federal Arbitration Act to make orders vacating arbitral awards appealable immediately, though nonfinal, 9 U.S.C. § 16(a)(1)(E); Perlman v. Swiss Bank Corp. Comprehensive Disability Protection Plan, 195 F.3d 975, 979-80 (7th Cir. 1999), unless the purpose of the remand was merely to enable the arbitrator to clarify his decision in order to set the stage for informed appellate review. Virgin Islands Housing Authority v. Coastal General Construction Services Corp., 27 F.3d 911, 913-14 (3d Cir. 1994). That was not the purpose here. Our error was harmless, however, because the appeal was in any event untimely. The district court had issued its order of remand on May 11, 1998, and the union did not file its notice of appeal until August 19, which was more than 30 days later.

We ended our order dismissing the union's appeal by saying that "the parties must proceed back through the federal courts to confirm or vacate the second award." This Delphic utterance left unexplained both how the union could challenge the district court's first decision, the decision vacating the arbitrator's award, which had been in the union's favor, and why the parties would have to file a brand-new suit (if that's what "proceed back" meant) in the district court. Let's take the second question first. There is no hard-and-fast rule, except in the Ninth Circuit, Shapiro v. Paradise Valley Unified School Dist. No. 69, 152 F.3d 1159 (9th Cir. 1998) (per curiam), and, by virtue of statute, in the social security area, 42 U.S.C. § 405(g); Shalala v. Schaefer, 509 U.S. 292 (1993); Richmond v. Chater, 94 F.3d 263, 266-70 (7th Cir. 1996); Raitport v. Callahan, 183 F.3d 101 (2d Cir. 1999) (per curiam); cf. Perlman v. Swiss Bank Corp. Comprehensive Disability Protection Plan, supra, 195 F.3d at 979-80, and perhaps other specialized areas with which we are not familiar, about when a remanding court loses jurisdiction over the case, thus necessitating a fresh proceeding of some sort after the remand to bring the issue back to that court. The presumption, however, is in favor of relinquishment. For when a court makes a limited remand unlikely to resolve the case but intended rather to assist the court in making its decision (a typical example being a remand to enable the district court to decide whether to enter a Rule 54(b) order to enable us to exercise appellate jurisdiction), the court will either state that it is retaining jurisdiction, e.g., Boyko v. Anderson, 185 F.3d 672 (7th Cir. 1999); Alexander v. Chicago Park District, 773 F.2d 850, 857 (7th Cir. 1985); United States v. Doe, 63 F.3d 121, 131 (2d Cir. 1995); Caterpillar, Inc. v. N.L.R.B., 138 F.3d 1105, 1107-08 (7th Cir. 1998) (chambers opinion), and cases cited there, or dismiss with leave to reinstate, which we treat as the equivalent of a stay. Baltimore Ohio Chicago Terminal R.R. v. Wisconsin Central Limited, 154 F.3d 404, 408 (7th Cir. 1998). The presumption was reinforced here by the district court's statement that it was "terminating case," suggesting that it was indeed relinquishing jurisdiction. No doubt the judge hoped that on remand the arbitrator would render an award that satisfied both parties or, more likely, would be solid enough to discourage a judicial challenge.

On remand, the arbitrator grudgingly ruled in favor of Jays, emphasizing that he disagreed with the district court's decision but was bowing to what he thought its implicit command to rule for the company. Naturally at this point the union wanted to get appellate review of the district court's initial decision, the decision vacating the arbitral award in the union's favor. It could have gotten this by filing a proceeding in the district court to set aside the second arbitral award on the ground that the court had erred in setting aside the first one. If the court was not persuaded and ruled in favor of Jays, the union could appeal to us and the appeal would bring up for review all nonmoot interlocutory orders issued by the district court, Lauer v. Apfel, 169 F.3d 489, 492 (7th Cir. 1999); LeBlang Motors, Ltd. v. Subaru of America, Inc., 148 F.3d 680, 689 (7th Cir. 1998); Librizzi v. Children's Memorial Medical Center, 134 F.3d 1302, 1305-06 (7th Cir. 1998); Keefe v. Prudential Property Casualty Ins. Co., 203 F.3d 218, 226 (3d Cir. 2000), including the order vacating the first arbitral award.

Granted, this is a slightly tricky point, since in our hypothetical example of the union's filing a fresh proceeding to set aside the award made by the arbitrator on remand we described as "interlocutory" an order technically issued in a prior case. But the usage is correct. When a case is appealed (here to the district court from the arbitrator rather than from the district court to this court) following a remand, there are two appellate proceedings but only one underlying litigation, and any interlocutory order in that litigation, even if it preceded the first appeal, is (within the limits of the doctrine of law of the case) open to review on the second appeal, unless it has become moot in the interim. Shearson Loeb Rhoades, Inc. v. Much, supra; cf. Reilly v. Waukesha County, 993 F.2d 1284, 1286-87 (7th Cir. 1993). The fact that an interlocutory order is appealable doesn't require that it be appealed, on pain of forfeiture if it is not. That would only encourage piecemeal litigation. A party can wait until the litigation is over and then bring a single appeal from the final judgment and challenge all nonmoot interlocutory orders, appealable or not, rendered along the way. Retired Chicago Police Ass'n v. City of Chicago, 7 F.3d 584, 608 (7th Cir. 1993); Chambers v. Ohio Dept. of Human Services, 145 F.3d 793, 796 (6th Cir. 1998); Ernst v. Child Youth Services, 108 F.3d 486, 492-93 (3d Cir. 1997); Sierra Club v. Robertson, 28 F.3d 753, 756 n. 3 (11th Cir. 1994).

Instead of taking up our invitation to file a new suit to challenge the second arbitral award, the union inexplicably moved the court in the original suit, the suit the district court had dismissed when it remanded the case to the arbitrator — a suit that consequently did not exist when the union filed its motion — to enter a "Rule 54" order. It is from the denial of that motion that the union appeals. The reference in the motion to "Rule 54" was presumably to Rule 54(b) of the Federal Rules of Civil Procedure, which permits a district court in certain circumstances to issue a final, and thus immediately appealable, judgment even though some piece of the case remains pending in the district court. E.g., Continental Casualty Co. v. Anderson Excavating Wrecking Co., 189 F.3d 512, 516-18 (7th Cir. 1999). The union must have thought it couldn't appeal the district court's order vacating the initial arbitral award without getting the court to make that order a Rule 54(b) judgment, because the order was interlocutory (and thus unappealable unless made a Rule 54(b) judgment); it must have thought the case was continuing in the district court awaiting a motion to vacate or confirm the second award. This was doubly wrong. The order vacating the arbitrator's first award, though indeed interlocutory, could, as we have emphasized, be appealed at the end of the entire litigation. And there was no pending proceeding in the district court, which had relinquished jurisdiction over the controversy when it sent the case back to the arbitrator. It is not surprising, therefore, that the district court's response to the Rule 54(b) motion was to deny it, on the ground that there was no case before the court. The motion was frivolous, and in no wise invited by our previous order. What is inexplicable, therefore, is why the union did not file a fresh challenge to the arbitrator's second award, which if confirmed by the district court could have been brought up to us for review by an appeal from that confirmation.

The denial of the union's motion for the entry of a Rule 54(b) judgment was correct, and is therefore

Affirmed.