  Wigner's seminal work on the Poincar\'e group revealed one of the fundamental
principles of quantum theory: symmetry groups are projectively represented. The
condensed-matter counterparts of the Poincar\'e group could be the spacetime
groups of periodically driven crystals or spacetime crystals featuring
spacetime periodicity. In this study, we establish the general theory of
projective spacetime symmetry algebras of spacetime crystals, and reveal their
intrinsic connections to gauge structures. As important applications, we
exhaustively classify (1,1)D projective symmetry algebras and systematically
construct spacetime lattice models for them all. Additionally, we present three
consequences of projective spacetime symmetry that surpass ordinary theory: the
electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet
crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows.
Our work provides both theoretical and experimental foundations to explore
novel physics protected by projective spacetime symmetry of spacetime crystals.
