Following this line, I am wondering about the following question.
Accreting pulsars in binary systems are usually thought to accrete from a prograde disk, so increasing their spin in the process.
But why is the disk supposed to be prograde?
Consider a binary system with a neutron star and a companion star. It is fair to assume that the two stars have had tidal interaction, such that their angular momentum is aligned. This is just saying that both stars rotate in the same direction as their orbital motion.
Now say that the companion is filling its Roche Lobe and steadily transferring mass into the gravitational potential of the neutron star. At this point there are two things we need to realize
Then, as the matter stream moves down the potential well, it will fall behind the neutron star (Coriolis effect), only to loop around and settle into an orbit (angular momentum). The disk then rotates in the same direction as the orbital motion, making it prograde with with neutron star rotation.
Note that this explanation is rather simple and by no means comprehensive. There may very well be peculiar situations in which a retrograde disk forms. I know that at least for black hole binaries there are some systems suggested to be retrograde.
For more details on mass transfer you can look at this file.
As far as I know all accretion disks are prograde. When the pulsar starts accreting mass from it's companion this mass is pulled to follow the pulsar's rotation.