During what type of process are the atoms that make up the substances involved actually rearranged?

Answer : During chemical changes only.

Explanation : A chemical change is usually when a compound or molecule undergoes a change which is normally irreversible and it generates a new product than it was before.

So, when we consider the process of making up of substances from atoms it makes the substance as an irreversible product.

Also a new product which is formed is the substance which is not at all similar to atoms.

Answer; 
Chemical change;

A chemical change is a change in which the type and the number of atoms that make up a substance are changed and rearranged. 

In chemical changes, the molecules or atoms in the reactants interact to form new substances.
On the other hand, in a physical change, like a state change or dissolving, no new substance is formed.