There are different groups operating under the name Tomiki Aikido or Shodokan Aikido. The koryu no kata of Tomiki Aikido are formalised by Hideo Ohba and his students. Kenji Tomiki was only involved in the 3rd and 4th set. There are 6 sets or koryu no kata. Most of the Shodokan people are focusing on 3rd and 4th set, of course there are exceptions. Some people say there is "Old School" and "New School". New School is Shodokan and JAA-groups which had ties in the past with Shodokan.
Old School are people who kept Tomiki's and Ohba's teaching inclusive the 6 koryu no kata. In the old school it is normal to practice both sides. I believe this is the same in new school, although some clubs maybe only do righthanded. According the grading syllabus you have to do it both sides.
At this moment some groups in the JAA are researching classical Kodokan Judo kata, because in the past the link between judo and aikido was much closer.
Just a thought,
The group I used to train with (a long time ago!) were BAA, who I am fairly sure were JAA affiliates, so I would guess you would class them as "new school". They also had a strong focus on shiai rather than kata, which would probably explain the right-hand bias.