If I have (say) a rod shaped neodymium magnet and wrap it with a coil, can I use the current in the coil to add to or subtract from the permanent magnet's field in a reversible manner?
Also, would the fields add in a linear manner or would saturation effects dominate?
Yes. At field strengths less than coercive force, superposition applies, and the total field is the sum of magnetization plus field strength.
Mind that most hard magnetic materials have low permeability, so it takes a lot of field strength to make a significant difference. But they generally also have high coercivity, i.e. it takes very high field strength to permanently change the set-in magnetization.
Saturation effects may or may not be an issue, but this will depend upon the pole pieces in the system, not the magnet itself. Basically, the magnet is already permanently saturated, which is also why / to say that its permeability is low.
Also mind that, under high enough instantaneous field strength, the magnet's magnetization will be affected. This is how they make them in the first place, after all. This is good reason to, for example, electronically limit peak current through the winding, or avoid exposing it to unlimited external transients. This is more of a concern in machines with iron pole pieces (Bsat > 1.2T) and ferrite magnets (Br ~ 0.4T), than with NdFeB (Br ~ 1.5T), and takes quite a lot of field strength for a bare magnet (no pole pieces) (ca. 105 A/m).
Oh, also keep in mind, changes in field strength induce eddy currents in the magnet, which will heat it up, and which act to oppose the self-inductance of the coil. (This also affects how quickly you can change the external field, and its distribution. See also: skin effect.) This is most significant for NdFeB and other metallic magnets; it's a non-issue for hard ferrites.
