It seems to me that sometimes moral rules are generalizations that guide moral thinking and action and sometimes they are universal/absolute principles applicable in all situations. The distinction between the two may well be related to the distinction between high-order and low-order moral judgements, I'm still trying to work through this distinction.
I need to clarify what I mean by universal/absolute. I think that morality is a human construct. As an atheist I do not think that my morality is derived from a supernatural being but that moral facts exist by virtue of human nature and human rationality.
We naturally tend to agree (perhaps it is fairly self evident or maybe it is through many years of debate and experience) that we are in favour of certain moral principles and against others. Principles such as justice, avoiding pain, putting right our mistakes, being grateful for assistance, treating everyone as valuable, equality etc. are the outcome of this process.
I think that as societies have 'matured', they have tended to adopt these moral principles more widely (e.g. abolishing slavery, treating men and women more equally etc.) but because human beings are flawed (our adrenal glands are too big) and politics is involved, there are always going to be conflicts and disagreements.
I do therefore think that it is a feature of a moral truth that it is universal. If it isn't universally true for all mankind then it isn't a moral truth but a political or cultural point of view.