Yes, from earth rotation the INS could figure out true north if the latitude is known. Or figure out the latitude if current heading is known. But normally it's aligned with a starting position from pilot input or GPS.
As I recall, it's a combination of things all happening at once, to include using last known location at shutdown, GPS inputs (if available) and inertial inputs.
There were various workarounds you had to use in the Navy, where the ship has not only usually moved from the last place the jet thought it was, but it's generally moving as the jet tries to re-align itself. GPS made things easier, and there wasn't a whole lot of thinking involved, but it always took the jet a bit more (or a lot more depending on model) thinking to get a stable platform underway.
If you are at an airport you will sometimes notice large signs giving Longitude and Latitude of the individual stands at an airport. These are used to give the initial position to the INS via the FMS. Of course these are now all built into the database these days so are only used (if at all) for gross error checking.