SC16IS752 not responding to SPI communication
The problem was that I had low level at the RST line, so SC16IS752 was in an endless reset.

Actually I had already measured the level at pin 2 and thought it was high. But it seems, that I measured an incorrect pin. After redoing all the checks and measurements, I noticed that actually the RST line is low. So I added a 1K pull up and SC16IS752 is now responding to what I'm sending on SPI.

Now that I know that, it seems to be plausible, that there is no internal pull-up on this line. You may want to control several chips with the reset signal, and when all of these would have internal pull-ups it would be a problem.

Thanks for your help.