Etymology: Left hand + work upon. This character has connotations of support / auxiliary. Thus, the original meaning was assist someone at work.
In time this came to mean only left hand, with 工 becoming redundant.
Hint: Take 工 to mean carpenter's square.
Mnemonic: LEFT HAND STEADIES CARPENTER'S SQUARE.
左折 (sasetsu) turn left
左側 (hidarigawa) left side
(Based upon Henshall, 1998)