I strop both diagonally and straight, and mostly diagonally. I start with linen, go to leather, and - even though this isn't usually done - then back to linen for about 2-3 passes.

The reason I go back to linen is because I noticed, using the HHT, that going back to linen for even a very small number of passes increases the efficiency of the HHT. I doubt I would have noticed this difference by the (most important) shave test.

I am using a vintage Illinois 6509 strop, and I love it! But, I have no historical awareness of what went on that strop thru the years. It doesn't appear pasted. But, I suspect there is a long buildup of oils and such. And, while that makes the strop work well, my theory at the moment is that the residue on the strop builds up on the edge and bevel during stropping.

My best guess is that returning to the linen for only a very small number of passes removes the small amount of buildup on the blade.

I am mentioning this because strop procedure was asked. I doubt the return to the linen strop is all that significant.