Is loop quantization in cosmology unique?

We reexamine the process of loop quantization for flat isotropic models in cosmology. In particular, we contrast different inequivalent "loop quantizations" of these simple models through their respective successes and limitations and assess whether they can lead to any viable physical description. We propose three simple requirements which any such admissible quantum model should satisfy: (i) independence from any auxiliary structure, such as a fiducial interval/cell introduced to define the phase space when integrating over noncompact manifolds; (ii) existence of a well defined classical limit, and (iii) a sensible "Planck scale" where quantum gravitational effects become manifest. We show that even when it may seem that one can have several possible loop quantizations, these physical requirements considerably narrow down the consistent choices. Apart from the so-called improved dynamics of loop quantum cosmology, none of the other available inequivalent loop quantizations pass the above tests, showing the limitations of lattice refinement models to approximate the homogeneous sector and loop modified quantum geometrodynamics. We conclude that amongst a large class of loop quantizations in isotropic cosmology, there is a unique consistent choice.