'Way back when, some twins only had one hydraulic pump - on the right engine - thus the 'start right first.' Four engined aircraft had two pumps, one each on engines two and three, but the entry door was on the left, thus start number three first, followed by two, one and four.
Some twin turboprops I've flown recommended alternating starts between the right and left engines, just to verify that the de-icer boot suction was working from that engine. If you intend to start any engine out of normal sequence, make sure the ground crew knows about it.
