Can ciphertext rotation be done without first generating a rotation index, or given a number of rotation steps, with steps greater than 0 rotating a few steps to the right and less than 0 rotating a few steps to the left.

Rotation by any index in SIMD schemes, i.e., BGV, BFV, or CKKS, requires a rotation evaluation key to transform the rotated ciphertext to the original secret key. So the keys for the given indices should pre-generated. Of course, one could reduce the number of keys, e.g., by generating a key for rotating by 1 and using it for many rotations, by 2, 3, 4 (which would be slower), but still at least one key needs to be generated.