In Banach space theory (closed) subspaces and quotient seem to play a symmetric role. However, since the behavior of subspaces is more intuitive, subspaces appear more frequently. E.g., the theory of Schauder basis.
In the case of operators between reflexive Banach spaces, the symmetry is perfect. Passing to dual spaces and conjugate operators, we get the quotient version of a subspace concept or result, and vice versa. Although this is not so clear in the non-reflexive case, many concepts and results formulated for subspaces admit a corresponding version for quotients:
Injective spaces (extension property) and surjective spaces (lifting property);
Strictly singular and strictly cosingular operators;
Subspace incomparability (no inf. dim. subspace of X is isomorphic to a subspace of Y) and quotient incomparability;
Q$_1$ Is there any general principle implying that the role of subspaces and quotients is non-symmetric?
Q$_2$ Is there any concrete subspace (quotient) result or concept that does not admit a corresponding quotient (subspace) version?
Q$_3$ Is there any concrete topic in which the theory for subspaces is intrinsically richer than the corresponding theory for quotients?