First tentative definition: axiom-initial ideologies are ideologies that start with one or more axioms and then explore the consequences of those and arrive at the conclusions from those axioms, whereas conclusion-initial ideologies start with one or more (usually multiple) conclusions and it simply ends there.

Examples of axiom-initial ideologies: Marxism, Libertarianism, Pantheism, Empiricism etc.
Examples of conclusion-initial ideologies: Christianity, Islam, Nazism, Emperor cults, etc.

In reality, however, it's very likely often (though not always) the case that ideologies do not usually neatly fit these descriptions, but an ideology may lean more into one or the other category, or an individual may lean more into one or the other category regardless of ideology. For instance, atheism may easily be either. Many ideologies may have two versions. Axiom-initial atheism starts with the premise of "nothing is divine" and then explores the consequences of that premise, whereas conclusion-initial atheism simply asserts that nothing is divine and then the individual moves on believing in that assertion without exploring the consequences of that statement. Ideologies like Marxism, Libertarianism and Pantheism have axioms that are way too broad and vague to be accepted as conclusions, and so, it's almost always the case that the consequences of their axioms are explored in depth. The same is not the case for Christianity, Islam etc. which start from the conclusions, and then the believer simply moves on without exploring the logical consequences of their beliefs. There are exceptions, of course, for instance with Christian and Islamic philosophers, who explored the logical consequences of their beliefs in depth, and those philosophers believe in axiom-initial [ideology], whereas those who don't question their beliefs believe in conclusion-initial [ideology]. However, in reality, these distinctions may not always be so binary.