Classic linear bilevel optimization assumes that the upper-level decision impacts only the right-hand side of the lower-level constraints, which guarantees a solution under mild conditions. We study existence of solutions for linear bilevel problems with rotations (LBPR, for short), i.e., problems for which the upper-level decision modifies the left-hand side coefficients of the linear follower constraints. Under a rather basic compactness and continuity assumption, if the LBPR is such that the follower is feasible for every leader decision, then relaxing all follower inequalities by any positive quantity guarantees a solution for the corresponding perturbed LBPR. Furthermore, we show that the problem’s domain can be partitioned into regions where the follower’s feasible-set mapping is continuous. Finally, we construct examples of LBPR with persistent non-attainment (irreparable) under prescribed perturbations of selected constraint coefficients, additive leader objectives, and nonnegative scalar right-hand-side relaxations. The simultaneous perturbation result keeps the leader domain, the fixed follower constraints, and the follower objective unchanged.