Traditionally to the off-side, the right, so that swords could be drawn without catching in the mane.
I'd always worked to keep manes to the right, and plaited to the right, until my last horse, who's mane was determined to sit on the left. When i let it to do it's own thing, it sat perfectly to the left and the world didn't come to an end.
I've never experienced it affecting temperament, however, I can imagine that if you force hair in the wrong direction it could cause soreness in the skin and muscle, irritating the horse until it gets used to the new side. I know what it's like to have a sore scalp when my long hair has been held against my natural shed or tightly in a braid and bun.
The best apple will be on the highest branch. Bidh an t-ubhal as fheàrr air a' mheangan as àirde. (Scots Gaelic proverb)