Although there is no federal rule expressly allowing motions to reconsider, parties file them anyway. Some evidence is that the PACER electronic filing system allows you to categorize a motion as one "for reconsideration."
A body of case law describes when motions to reconsider are appropriate. Following is a section from a brief (not mine) that quotes from a number of Illinois cases on the issue:
A motion to reconsider is available to correct manifest errors of law and fact, or to present newly discovered evidence. Wolves v. National Railroad Passenger Corp., 1990 WL 84519, at *1 (N.D. Ill. 1990) (not reported in F. Supp.) (citing Publisher's Resource Inc. v. Walker-David Publications, Inc., 726 F.2d 557, 561 (7th Cir. 1985)). A motion to reconsider should be used neither to introduce new evidence that was available during original consideration, nor to introduce new legal theories. Calderon v. Reno, 56 F.Supp.2d 997, 998 (N.D. Ill. 1999) (citing Publisher's Resource, 762 F.2d at 561)).
A motion to reconsider a remand order is proper "where the court has patently misunderstood a party, or has made a decision outside the adversarial issues presented to the Court by the parties, which made an error not of reasoning but of apprehension." Calderon, 56 F.Supp. 2d at 999 (citing Bank of Waunake v. Rochester Cheese Sales, Inc., 906 F.2d 1185, 1191 (7th Cir. 1990)). "A further basis for a motion to reconsider would be a controlling or significant change in the law or facts since the submission of the issue to the Court." State Bank of India v. Commercial Steel Corp., 2001 WL 423001, at *2 (N.D.Ill. 2001) (not reported in F.Supp.2d). "Such problems rarely arise and the motion to reconsider should be equally rare." State Bank of India v. Commercial Steel Corp., 2001 WL 423001, at *2 (N.D.Ill. 2001) (not reported in F.Supp.2d) (citing Rounds v. City of Chicago, No. 94 C. 1708, 1996 WL 99408, at *1 (N.D. Ill. March 5, 1996) (citations omitted), aff'd, 95 F.3d 1154 (7th Cir. 1996)).
That's a pretty good description of the circumstances under which a motion to reconsider might be appropriate, and I wanted to bookmark it here for future use.