  It is possible to implement a certain form of modified gravity inspired by
loop quantization through non-bijective canonical transformations. The
canonical nature might suggest that such modifications are guaranteed to
preserve general covariance. Here, however, we show that a dedicated space-time
analysis is still required, even in the case of a bijective canonical
transformation. In addition, a complete global analysis is presented for a
recent proposal of a non-bijective transformation, showing that it does not
preserve general covariance and that the only novel physical effect introduced
by the modification is the presence of certain time-reversal hypersurfaces
between classical space-time regions. These results provide further insights
into the physical interpretation of modified dynamics in models of loop quantum
gravity.
