1839, "arrangement (of a building, etc.) to face east or any other specified direction," noun of action from orient (v.). Meaning "process of determining the points of the compass is by 1868, hence the extended sense of "action of determining one's mental bearings," with reference to new ideas, etc. (1870). Meaning "introduction to a situation" is from 1942. Sense of "the position or arrangement (of something) relative to the points of the compass" is from 1875. Related: Orientational.
1849, "to turn or cause to turn toward the east," a back-formation from orientation. Intransitive sense of "assume an easterly direction" is by 1850. Figurative meaning "take one's proper bearings mentally" is by 1866. Related: Orientated; orientating.
The Latin sense shift in auster, if it is indeed the same word other Indo-European languages use for "east," for which Latin uses oriens (see Orient (n.)), perhaps is based on a false assumption about the orientation of the Italian peninsula, "with shift through 'southeast' explained by the diagonal position of the axis of Italy" [Buck]; see Walde, Alois, "Lateinisches etymologisches Wörterbuch," 3rd. ed., vol. I, p.87; Ernout, Alfred, and Meillet, Alfred, "Dictionnaire étymologique de la langue latine," 2nd. ed., p.94. Or perhaps the connection is more ancient, and from PIE root *aus- "to shine," source of aurora, which also produces words for "burning," with reference to the "hot" south wind that blows into Italy. Thus auster "(hot) south wind," metaphorically extended to "south."