Congruence mod 5 partitions the integers into exactly five classes, while "less than or equal to" partitions nothing. Both are reflexive and transitive. What does the third property decide?

B. Symmetry versus antisymmetry: one groups, the other arranges.

Congruence is symmetric, so relatedness is mutual and a group of mutually related elements has no internal direction, a clump. Ordering is antisymmetric, so the relation runs one way and arranges rather than groups. Reflexive+symmetric+transitive gives a partition; reflexive+antisymmetric+transitive gives a hierarchy. Which you have decides what you can do.