Summary
In many fields there's a divide between intuitive, axiomatic principles and rational, derived principles. Ethics contains these two sides as well. Our ethical conclusions are a balance between intuitive reactions and rational conclusions, with feedback in both directions.
Two modes of understanding
In math and other fields of rational inquiry there are two degrees of understanding. The first is intuitive knowledge. Whether because we experience something directly or because we conceptualize it through example, intuitive facts just feel right. For example, it seems obvious that the angles opposite the equal sides of an isosceles triangle will be the same. At times, concepts are not immediately intuitive but can become so through analogy, such as the realization that the total resistance of two wires connected in parallel is less than the resistance of each wire individually because one can draw a comparison to the amount of water that can flow simultaneously through two pipes rather than one. The second mode of understanding is formal proof. We may intuitively know that all of the angles of an equilateral triangle are equal, but we also must prove it to be sure. Math and science abound with examples in which counterintuitive ideas proved correct: heliocentrism, relativity, and quantum mechanics, etc. Complete understanding of something generally requires both modes of thought.
The same process of understanding applies to morality. Ethical questions should not be solely decided on the basis of pure intuition or on the basis of pure reason. Indeed, it's hard to even imagine what it would look like for something to be decided purely on the basis of reason. At some point, one needs to establish why things matter, and it's hard to imagine how this could be done without emotion or some other motivating force.
Axioms
The basis of morality must always be intuition: This grounds and gives meaning to any reasoning that one hopes to undertake. For example, one must intuitively establish that "suffering is bad" before the rational realization that "this action will reduce net suffering" has any moral significance. To use Toulmin's terminology of argument, one must establish a warrant before an audience will make the leap from one's grounds to one's claim. At the foundational level, such warrants most plausibly arise from intuition.
In the same way, mathematics arises from certain core axioms that one takes for granted. In the same way that two people can disagree about the core values of ethics, two people may take for granted different core axioms in math and arrive at two different theories (take, for example, Euclidean and non-Euclidean geometry).
From Methods of Ethics, "Preface to the Sixth Edition" (1901):
Thus, in spite of my early aversion to Intuitional Ethics [...], I was forced to recognise the need of a fundamental ethical intuition.
The utilitarian method [...] could not, it seemed to me, be made coherent and harmonious without this fundamental intuition. [...]
And I had myself become, as I had to admit to myself, an Intuitionist to a certain extent. For the supreme rule of aiming at the general happiness, as I had come to see, must rest on a fundamental moral intuition, if I was to recognise it as binding at all. And in reading the writings of the earlier English Intuitionists, More and Clarke, I found the axiom I required for my Utilitarianism [That a rational agent is bound to aim at Universal Happiness], in one form or another, holding a prominent place [...].
Deductions
Once basic assumptions have been established, we generally let reason take its course. While intuitive mathematical conclusions do often turn out to be correct by logical proof, there are some instances in which that is not the case; sometimes theorems that are clearly proved from basic axioms turn out to be counterintuitive (e.g, the Banach-Tarski paradox).
Sometimes we should do the same in ethics. Beyond the most basic, axiomatic expressions of value, pure intuition can become fuzzy, capricious, and self-contradictory. Reason can guide our emotional energies toward those decisions that genuinely are most in line with our fundamental intuitions.
Two-way street
That said, there are times when intuitions may override deductions from reason, if a conclusion is too repugnant for us to countenance. I tend to bite a fair number of bullets, but I acknowledge that sometimes it can make sense to reject the premises that lead to an unacceptable result. This is not a violation of rationality, because the axioms were never certain to begin with. Indeed, even with the Banach-Tarski paradox, some consider that result a refutation of the axiom of choice.
Our introspection has limits. Moreover, our intuitions are a complex mixture of many impulses, and they don't agree even in principle. Deciding which intuitions to accept and which to eliminate is a matter for our hearts to decide after long reflection, not based on principles that may seem sensible at first glance.
Further reading
Compare to Joshua Greene's "dual-process theory" in moral psychology.