Why does a transformation on $p$ imply the inverse transformation on $q$ in phase space?

The author of this statistics paper aims for transformations $pmapsto p',,qmapsto q'$ that preserve Hamiltonian geometry. As far as I can tell from a quick look at Hamiltonian Monte Carlo this means he wants to preserve the energies expressed as $$ H=frac{1}{2}boldsymbol p^top M^{-1}boldsymbol p,,quad T=frac{1}{2}dot{boldsymbol q}^top Mdot{boldsymbol q},. $$ This looks quite familar to a physicist. Now take the square root of $M$ and $M^{-1},.$ Clearly, the invariance of $H$ and $T$ under the above transformations is compatible with $boldsymbol p'=sqrt{M^{-1}}boldsymbol p$ and $dot{boldsymbol q}'=sqrt{M}dot{boldsymbol q},$ and assuming that $M$ is constant we can undo the time derivative of the $dot{boldsymbol q}$ relation.