Starting from two Lagrangian immersions and a Legendre curve $${\tilde{\gamma}(t)}$$ in $${\mathbb{S}^3(1)}$$ $$({\rm or\,in}\,{\mathbb{H}_1^3(-1)})$$ , it is possible to construct a new Lagrangian immersion in $${\mathbb{CP}^n(4)}$$ $$({\rm or\,in}\,{\mathbb{CH}^n(-4)})$$ , which is called a warped product Lagrangian immersion. When $${\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i(- \frac{r_1}{r_2}at)})}$$ $$({\rm or}\,{\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i( \frac{r_1}{r_2}at)})})$$ , where r 1, r 2, and a are positive constants with $${r_1^2+r_2^2=1}$$ $$({\rm or}\,{-r_1^2+r_2^2=-1})$$ , we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of $${\mathbb{CP}^n(4)}$$ or $${\mathbb{CH}^n(-4)}$$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations.