Eight kinds of equivalence classes (five of which are new) within the family $L(n)$ of finite loops on $n$ elements $(n\leq6)$ are considered. The classes arise by combining the operations of isotopy over $L(n)$ (with some of its specializations) and loop-parastrophy (parastrophy followed by a special isotopy, which returns the image to $L(n)$). The used isotopies are triples of permutations of the ground-set (applied successively to rows, columns and elements of the associated Cayley table) which map $L(n)$ onto $L(n)$. Classical isotopy and isomorphic classes correspond to the triples of the form $(p,q,r)$ and $(p,p,p)$ respectively. Three new natural kinds of interclasses, denoted as $C$-, $R$- and $E$-classes, correspond to the triples of the form $(q,p,p)$, $(p,q,p)$ and $(p,p,q)$ respectively. The combinations ``isotopy over $L(n)$ + loop-parastrophy'' and ``isomorphism loop-parastrophy'' lead to the classical main classes and to a new kind of classes, denoted as II-classes. Finally, a new kind of classes, called paras-trophic closures, corresponds to the transitive closure of the loop-parastrophy operator. Cardinalities, intersections and dualities for all the eight kinds of equivalence classes of loops are completely determined for $n\leq6$. In addition, the following theorem, related to classical isomorphic, isotopy and main classes, is proved by using the new II-classes: All the isotopy classes within a main class have the same family of cardinalities of their included isomorphic classes.