Background

I have been reading about Symmetry operators and had a straight forward question. When we have a normal operator like momentum or Hamiltonian we know it acts over a wavefunction as $\hat{H}\psi$ or $\hat p\psi$. But when we talk about Symmetry Operators, everywhere I see it being operated along with its inverse like $U\psi U^{-1}$ where U is any of Symmetry operator.

Questions

Q What is the meaning of being operated with along with its inverse?

Q They also talk about the sense of even-ness and oddness whenever they talk about symmetry operator acting in this way. Is the same even-ness odd-ness concept applicable to say normal operator of Hamiltonian?