We study the homomorphism-homogeneity of normal bands \(B=[Y;B_{\alpha };\psi _{\alpha ,\beta }]\) (that is, a spined product of a left normal band L and a right normal band R). We show that if B is homomorphism-homogeneous, then one of the following conditions holds: (i) B is image-trivial; (ii) B is surjective; (iii) L is image-trivial (resp. surjective) and R is surjective (resp. image-trivial). Furthermore, we show that any iso-normal band is homomorphism-homogeneous if and only if its structure semilattice is homomorphism-homogeneous. Consequently, a non-image-trivial injective normal band B is homomorphism-homogeneous if and only if B is iso-normal and its structure semilattice is homomorphism-homogeneous. We also classify homomorphism-homogeneous image-trivial normal bands whose structure semilattices are trees. It is shown that any such normal band is homomorphism-homogeneous if and only if it has trivial minimum-collapsing, which is a restriction on the size of the \(B_\alpha \) . Consequently, an image-trivial normal band in which \(R_\alpha \) is non-trivial for all non-minimal elements \(\alpha \in Y\) is homomorphism-homogeneous if and only if Y is a tree and B has trivial minimum-collapsing.
