Thinking out loud on my issue I'm posting the evolution of it. Something quite strange is happening. I've used code that uses Arduino's SPI to read data from shift registers and the bit shift problem remained, but I accidentally wired a switch on pin 10 of the shift register, which is supposed to transmit data from Arduino's MISO to the daisy chained chips, and that switch was expressed in the first bit of that byte. So what actually happens is that the chip's pins have been physically shifted by one...you can check the chip's datasheet here http://www.ti.com/lit/ds/symlink/cd74hc165.pdf.
So now pin 10 (DS) works in the place of pin 11 (D0), pin 11 in place of 12 (D1), D1 in place of D2 etc. and D7 doesn't do anything if it's the first chip of more daisy chained chips. If the chips is the second one, then pin D7 affects the first bit of the first chip's byte, if the chip is the third, D7 affects the first bit of the second chip's byte etc. some kind of strange wrap around.
Also, when using a switch on pin 10, the chips following chips drop all their pins low (with pullup resistors)! If using a switch on pin 10 of the last chip, nothing weird happens in the other chips.
Anyway, I have no idea at all why all this is happening, but I can compensate for my project's needs, so more or less, problem solved.
BTW, is there any chance I'm damaging the shift registers by using pin 10 (DS) this way?