Sign of a classical field theory Lagrangian

Question

In classical field theory the action of massive real scalar field on a curved background reads: $$I = \int_M \sqrt{-g}\,d^4x\,\left[ - \frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m\phi^2 \right]\; ,$$ where the signature of the space-time is (-,+,+,+) and $M$ is the space-time (oriented) manifold. If we consider an orientation reversing diffeomorphism $\psi\, : M\rightarrow -M$ such that $y=\psi (x)$, and then we insert the change of variables $\psi^{-1}$, i.e. $x=x(y)$, into the action above, as far as I know I should obtain the following equivalent action $\tilde{I}$, namely $$\tilde{I} = \int_{-M} \sqrt{-\gamma}\,d^4y\,\left[ \frac{1}{2}\gamma^{\mu\nu}\tilde{\partial}_{\mu}\phi\tilde{\partial}_{\nu}\phi-\frac{1}{2}m\phi^2 \right]\; ,$$ where now the signature is (+,-,-,-). Indeed, by reversing the orientation of the manifold, the signature of the space-time would change its sign. Here is my question: if the volume form would change as it should because of the orientation-reversing map, why does the mass term remain unchanged? From my point of view, the only acceptable answer is that the integration for the mass term should be understood as a "measure". Am I correct? Thank you in advance for your contributions.