So rotational spectroscopy is something one can spend a lot of time talking about so I'll try to keep this brief. Although this isn't a complete answer, it was too long for a comment, but hopefully this sheds some light on the topic.

For rotational transitions, one of the selection rules tells us that in order for EMR to induce a transition in the rotational energy levels, a permeant dipole moment must exist in the molecule (i.e. a single atom wont be observed in rotational spectra, nor would a molecule such as $H_{2}$).

The rotational symmetry number is a variable that is used to account for the amount of rotations one can perform on a molecule that would make it indistinguishable from the starting point (i.e. a 90 degree rotational of $PCl_{5}$ along the principal axis can be performed three times before returning each atom to its original position in space, however, each rotation produces an identical picture of the molecule each time).

Moment of inertia can be calculated from the rotational constant, however, in practice, one would usually calculate the moment of inertia with computational softwares and then use that to approximate the rotational constants which are employed to predict the energies and allowed transitions for the rotations of a molecule.

This is a link to an appendix with the equations used for calculating the moment of inertia in molecules based on the geometry (e.g. symmetric rotor, linear)

Here is an article that further discusses symmetry numbers and parition functions

This one talks about rotational partition functions of diatomic gases

This article talks about transition moment integrals and selection rules

So rotational spectroscopy is something one can spend a lot of time talking about so I'll try to keep this brief. Although this isn't a complete answer, it was too long for a comment, but hopefully this sheds some light on the topic.

For rotational transitions, one of the selection rules tells us that in order for EMR to induce a transition in the rotational energy levels, a permanent dipole moment must exist in the molecule (i.e. a single atom won't be observed in rotational spectra, nor would a molecule such as $\ce{H_2}$).

The rotational symmetry number is a variable that is used to account for the amount of rotations one can perform on a molecule that would make it indistinguishable from the starting point (i.e. a 90 degree rotational of $\ce{PCl_5}$ along the principal axis can be performed three times before returning each atom to its original position in space, however, each rotation produces an identical picture of the molecule each time).

Moment of inertia can be calculated from the rotational constant, however, in practice, one would usually calculate the moment of inertia with computational software and then use that to approximate the rotational constants which are employed to predict the energies and allowed transitions for the rotations of a molecule.

This is a link to an appendix with the equations used for calculating the moment of inertia in molecules based on the geometry (e.g. symmetric rotor, linear)

Here is an article that further discusses symmetry numbers and partition functions.

This one talks about rotational partition functions of diatomic gases.

This article talks about transition moment integrals and selection rules.