Isomorphisms between quotient algebras of a given algebra form, under an appropriate multiplication, an inverse semigroup here called the inverse semigroup of bicongruences. This paper determines those normal bands whose inverse semigroups of bicongruences belong to special classes of inverse semigroups, and finds conditions on the structure semilattices, rectangular band components and linking homomorphisms of two normal bands with isomorphic inverse semigroups of bicongruences, under which the bands themselves are isomorphic or anti-isomorphic, or related in some looser manner. Some results are also given for the wider class of general bands.