Yes, The old method used to be to allow the new canes to grow to one side, say to the left and the old ones to the right. Then obviously it swaps from year to year. Don't think it is as important with thornless ones.
Once the old canes have finished fruiting, they may be removed down to ground level and the new canes tied in. They fruit better on new wood.

Thanks O, that's just what I wanted to hear.

I remember this question from when you bought it - its doing well isn't it!

2 types of blackberry floricane or primocane. Primocane types crop on any new cane in the same year usually twice, 1st time late summer then you cut it down to a few feet and it crops the following year on that same cane growth. In my experience in the UK primocane is a big waste of time growing, fruit is not that great tastewise too.

The type most people grow is what you have now floricane where the roots are perennial, but the canes biennial meaning after 1 year once they have flowered and fruited they are dead and new canes come from the crown or sucker sometimes off roots. You leave them to grow for the next year removing any root suckers( thornless usually don't) which stray too wide. I grow a Swiss blackberry called Navaho Summerlong which gives an incredible amount of tasty juicy fruit(8 kg +) and suckers freely from its roots once established- cut them out- free plants for other locations. Bonus is upright, no need to tie in canes as they are very upright!