While you're right that transversus is masculine and transversarium is neuter, the latter is not genitive, but these are actually two different words.
The first can be broken into three parts: trans-, vers-, and -us.
The second, however, into four: trans-, vers-, -ari-, -um.
The first is a pure past participle from transverto ("to turn away"). Here ...
There are two distinct words here:
The noun vertebra.
The adjective vertebralis, "related to vertebra".
The adjective is derived from the noun, and both the noun and the adjective have various different forms.
This is an important starting point and helps make sense of such Latin terms.
Adding the prefix inter- to the adjective turns it into a ...