  For classical gravitational systems the lapse function and the shift vector
are usually determined by imposing appropriate gauge fixing conditions and then
demanding their preservation with respect to the dynamics generated by a
canonical Hamiltonian. Effective descriptions encoding quantum geometric
effects motivated by loop quantum gravity for symmetry reduced models are often
captured by polymerization of connection (or related) variables in gauge fixing
conditions as well as constraints. Usually, one chooses the same form of
polymerization in both cases. A pertinent question is if the dynamical
stability of the effective gauge fixing conditions under the effective dynamics
generated by the polymerized canonical Hamiltonian is provided by the lapse
function and the shift vector obtained from the polymerization of their
classical counterparts. If this is the case, then we say that gauge fixing and
polymerization commute. In this manuscript we investigate these issues and
obtain consistency conditions for the commutativity of gauge fixing and
polymerization. Our analysis shows that such a commutativity occurs in rather
special situations and reveals pitfalls in making seemingly well motivated
choices which turn out to be inconsistent with the effective dynamics. We
illustrate these findings via examples of symmetry reduced models in the loop
quantization of the Schwarzschild interior and Lema\^{\i}tre-Tolman-Bondi (LTB)
spacetimes and report the non-commutativity of gauge fixing and polymerization,
and inherent limitations of some choices made in the literature with a
consistent effective dynamics.
