Conventionally latitude precedes longitude.

analysis.cartography.get_lat_lon_grids(cube) returns lats, lons
analysis.cartography.unrotate_pole(...) returns lons, lats

More generally:
grep -r 'lat._lon' * | wc -l
1668
grep -r 'lon._lat' * | wc -l
206

Does this need to be consistent, or return a dictionary {lat: y, lon: x}?

Currently the Iris documentation does not state which order the coords will be returned for rotate_pole and unrotate_pole.