Yes, intro- is a Latin prefix meaning "to the inside" while extra- means "to the outside". The "-vert" is from the Latin verb "vertere", meaning "to turn".

Adding Latin prefixes to "vertere" leads to such words as "to avert" (a- from Latin "ab-", meaning "from"), "to evert" (from "ex-" which means "out"; compare "extra-" above), "to invert" (from "in-" meaning "into"; compare "intro-" above); "reverse" (an adjective meaning 'opposite' from "re-" meaning "back"); "transverse" (an adjective meaning 'crosswise' from "trans-" meaning "across"); and so forth.

It's useful to compare "in-" vs "intro-" and "ex-" vs "extra-". The "-tra-" and "-tro-" infixes are essentially the same, but Latin writers and speakers inflected them historically. English inherited this in such words as "introduction", "extradition", "introspection", "extraordinary", and "extravagant".

Finally in modern biological and medicinal jargon it is common to use "intra-" (with an "a") as "within": "intracellular" (within cells; compare with "intercellular", "between cells"), "intradural" (within the dura mater in the spinal cord), "intramolecular" (e.g. internal forces holding together the shape of a molecule, or the atoms a molecule comprises), and so forth. Outside those technical settings, "intra-" is exceedingly rare. One example is "intramural" ("within the walls", usually used to describe a competition between different teams of students at a single school, however the word has use in medicine as well, the walls being those of a cell, blood vessel, or other hollow organ, and the medical use likely came first).