The talk will focus on the question of whether existing symplectic methods can distinguish pseudo-rotations from rotations (i.e., elements of Hamiltonian circle actions). For the projective plane, in many instances, but not always, the answer is negative. Namely, for virtually every pseudo-rotation there exists a unique rotation with precisely the same fixed-point data. However, the hypothetical exceptions—ghost pseudo-rotations—suggest that the relation between the two classes of maps might be much weaker than previously thought, possibly leading to some unexpected consequences. This is based on joint work with Viktor Ginzburg.