No. The base (aka the solids container) of the Air Head is manufactured with the handle on one side or the other. The handle is actually slightly offset and doesn’t run directly across the centre. It is possible to purchase another base with a different handle position if needed.
By default, the Air Head is supplied with the turning handle on the right, as seen when you are standing in front of the toilet, looking at it (so on the left when you are sat on the toilet).