There's a page on stropping in the library, with a diagram near the top showing the x-stroke:
here. The blade should follow the opposite diagonal on the way back (ie if you do bottom left - top right, coming back is top left - bottom right.)
As far as I know, the material of the strop affects the 'draw' - that's how much friction you feel using it. Hard shiny ones have less draw than soft suede-y ones. I think it's a matter of personal preference which you use. (Although my physics head says that a higher friction/draw should do whatever it is a strop does quicker. I stand to be corrected on that :) ).