I would throw the dice in such a way that I wouldn't have to manually establish the order.

The rules presented here ('pick from left to right, from top to bottom when left/right is unclear') will cause unintentional choices: there will be edge cases where two dice come resting in such a way that both rules could be used, but result in opposite order.

You might unconciously order these pairs in a non-random way (say, ascending in number).

Not sure if the bias could be statistically relevant.

Throw only a single dice (no ordering issue), use multiple coloured dice in parallel (and always use the colours in the same order), or use a shaking cup which blindly (!) orders the dice in some way.