  We propose a generic mechanism to distinguish between two different phases in
systems exhibiting excited-state quantum phase transitions (ESQPTs). Inspired
by the classical constant energy surfaces of quantum collective systems, we
find an operator, $\hat{\mathcal{C}}$, which is a constant of motion only in
one of these two phases. This means that thermodynamics is qualitatively
different in both phases. In the phase having this constant of motion,
long-time averages of physical observables crucially depend on its expected
value. We also show that such an operator acts like a discrete symmetry in the
corresponding phase. This fact explains one of the most common features of
ESQPTs -- the change from degenerate doublets to non-degenerate energy levels
at the critical energy of the transitions in systems having an additional
discrete symmetry, like parity. Our arguments apply generally for a large class
of systems, and we provide numerical examples in two paradigmatic models with
one and two degrees of freedom.
