This is a common source of disagreement. Both “orient” and “orientate” are verbs meaning to align or position yourself; to work out where you are within a particular situation or environment. The origin of both words is the same : the Latin word oriens meaning “rising” and “east”, because of the rising sun.
Orient as a noun means the countries of the East, especially those of east Asia. Strictly speaking, then, to orient/orientate yourself means to align yourself to the east, although the verb now has the general sense of “to position yourself”.
In the UK, it is more common for people to say “orientate” whereas in the US, “orient” is more common. Writers in both countries sometimes bemoan the usage of the alternative word. In fact, both words are acceptable according to the dictionaries.
The Oxford English Dictionary and the Merriam-Webster Dictionary list “orient” and “orientate” as verbs meaning the same thing. Which one you choose to use really just comes down to local preference. To a UK reader, “orient” may well sound non-standard, whereas “orientate” may sound clumsy to a US reader. Other parts of the world will have their own preferences. The key thing to remember is that both forms of the verb are generally acceptable.
As an aside, the opposite of Orient (the noun) is “Occident” : the countries of the West. There is, however, no equivalent verb. You can neither “occident” nor “occidentate” yourself. The closest verb is occidentalize, meaning to conform to western ideas or customs.