Why is variable rotation uncommon in cryptographic primitives? Most cryptographic primitives I've seen rotate by a constant. RC5 did something different though:



For a word size $w=2^n$, you can take the last $n$ bits of a value as a rotation amount. There's more sophisticated variants like that used in RC6.

To give these values a name: $A \lll B$.

Some good things:


Rotations are fast
Provides some diffusion
Has some nonlinear properties
Can be strong when combined with other operations


Some bad things:


Parity of $A$ is preserved
Differences in $A$ are preserved
Only a few bits in $B$ are used
Differences in $B$ may not have an effect (or have little effect)
Less differences if $A$ is mostly (or all) 0s or 1s
Less differences with alignment in $A$
Multiple variable rotations are not proportionally better


There's probably some more subtle weaknesses that make it more susceptible to differential cryptanalysis, but nothing that totally breaks it I think, or RC5 would be unusable.
One issue is that data-dependent rotations (such as you describe) is patented by RSA data security (or, at least, was, the patent may have expired).  RC5 and RC6 was created by the holder of this patent, however such a patent could be enforced against someone else, and so people have shied away from it.

More minor issues would include:


It is likely to take variable time on lower-end CPUs, and hence potentially vulnerable to time-based side channel attacks.
Such variable rotates are moderately expensive in hardware (when they need to be implemented in constant time); yes, you can construct a barrel shifter, but that takes a number of gates and interconnects.