Compatibilists assume the truth in some sense of determinism and the truth in some sense of freedom. Their view is not 'independent of whether or not determinism is true'. Compatibilism assumes the truth of determinism but aims to persuade us that the truth of determinism is compatible with the reality of freedom - which here means free will. This looks to be some task, but that's the aim.
Take determinism to be the view that for any event or state of affairs, E, (including actions) there are preceding conditions, C, of which the occurrence is sufficient for E. In other words, given C, E will occur.
This formulation won't satisfy everybody but the main issue involved in the debate about compatibilism does not turn, I think, on the precise definition of determinism. What compatibilists hold is that our actions can be determined and yet still be free. They are able to reconcile freedom and determinism, not by offering a reconceptualisation of determinism but by offering a reconceptualisation of freedom.
For the compatibilist, I am free if and only if certain constraints on my actions are absent. Typically these constraints are coercion and compulsion imposed by another agent.
Assume that my actions are determined - say, causally determined by my genetic inheritance. I have a particular character and a pattern of motivation and intentions, all of which are determined and produce my actions. My genetic inheritance, C1, is sufficient for E1, my particular character and pattern of motivation and intentions. (This is a crude picture but it will serve to make the main point.) When I act I could not have done otherwise - so what? So long as my actions are the products of E1, without interference by others in the way of coercion or constraint, I act freely. Once coercion or constraint is introduced, I am unfree.
So free will is opposed not to determinism but to coercion or constraint. This is a perfectly intelligible position. The only trouble is that it reconceptualises freedom and renders the traditional problem of free will and determinism unrecognisable.
As the problem of free will has been traditionally understood, free will and determinism cannot both be true. They are incompatible because (on one standard approach) the human agent has a freedom of choice regardless of all and any preceding conditions, C.
While this formulation of free will has the merit of being traditional, it has the drawback that it seems to make human actions merely random (so not ascribably 'our' actions at all but merely things that happen) or to involve a non-empirical self, outside the realm of determinism (since it is not subject to determinism) but able to act within it. Is such a self a coherent possibility?
If compatibilism redefines freedom and sidesteps the traditional debate between free will and determinism, incompatibilism of the kind just described leaves the explanation of free action at least not transparently coherent. But this is to raise issues beyond the immediate range of your question.
For the record I have never been able to reach a stable view about free will and determinism. But this is autobiography, not philosophy.