What does it mean to say that a curve in state space describes change with respect to time, as opposed to space or any other parameter? What does it mean to say it's time is asymmetric? Inspired by the Wigner-Bargmann analysis of the Poincaré group, I discuss a general framework for understanding the meaning of time evolution and temporal symmetry in terms of the representation of a semigroup that includes "time translations", amongst the automorphisms of a state space. I discuss the structuralist and functionalist philosophical underpinnings of this view, and show how time reversal, parity, matter-antimatter exchange, and CPT are best viewed as extensions of a representation of continuous symmetries, whose existence is sensitive to the underlying structure of state space. I conclude with some comments on how an arrow of time can be defined in this framework, as well as prospects for such an arrow in the context of gravitation.