Looks like 'Purple Plena Elegans' although a bit early for that to be in bloom (it is a viticella/late blooming, pruning group 3 type). Flower size should increase once established in the ground but they are on the small end.

I also think Purple Pleno. Mine is more smoky purple and a prolific bloomer.

Thank you gardengal48 and mnwsgal. The above pic was taken on June 12 last year, and the plant is in nearly full bloom now. Would that make it unlikely to be 'purple pleno' ? However, the pictures that I was able to find of 'purple pleno' seem just like it.

'Purple Plena Elegans' is pretty much the only one it could be...there aren't many that fully double with tiny flowers. The other options are rare and different colored.

Thanks buyorsell888, I appreciate the confirmation!