There exist Casimir invariants of the Galilean group which commute with all the generators of the group. They are, of course, Galilean scalars (i.e., scalars under space and time translations, rotations, and Galilean boosts). Two Casimir invariants of the Galilean group in 3+1 D are $$ME-P^2/2$$ and $$W^2~~{\rm where}~~{\vec W}=M{\vec L}+{\vec P}\times{\vec K}.$$
What do these invariants mean physically like for the Poincare group, the Casimir invariants represent the invariant properties of the particle: its mass and spin.
