It is merely a visual change.
The LeftHanded bit tag determines whether or not the mob's mainhand is its left or right hand (when set to 1, it is left-handed). The equipment held within HandItems remains the same. That is, the first record is the mainhand and the second is the offhand (in both cases, the stone is the mainhand).
/summon Zombie ~ ~1 ~ {HandItems:[{id:"minecraft:stone",Count:1b},{id:"minecraft:dirt",Count:1b}],LeftHanded:1b}/summon Zombie ~ ~1 ~ {HandItems:[{id:"minecraft:stone",Count:1b},{id:"minecraft:dirt",Count:1b}],LeftHanded:0b}