Two Hamiltonian cycles $$C_1=\langle u_0,u_1,u_2,...,u_{n-1},u_0 \rangle $$C1=źu0,u1,u2,...,un-1,u0ź and $$C_2=\langle v_0,v_1,v_2,...,v_{n-1},v_0 \rangle $$C2=źv0,v1,v2,...,vn-1,v0ź of a graph G are independent starting at $$u_0$$u0 if $$u_0=v_0, u_i\ne v_i$$u0=v0,uiźvi for all $$1\le i\le n-1$$1≤i≤n-1. A set of Hamiltonian cycles C of G are k-mutually independent starting at vertex u if any two different Hamiltonian cycles of C are independent starting at u and $$|C| = k$$|C|=k. The mutually independent Hamiltonianicity of graph G is the maximum integer k, such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles starting at u, denoted by IHC($$G)=k$$G)=k. The Cartesian product of graphs G and H, written by $$G \times H$$G×H, is the graph with vertex set $$V(G) \times V(H)$$V(G)×V(H) specified by putting (u, v) adjacent to $$(u', v')$$(uź,vź) if and only if $$(1)\;u = u'$$(1)u=uź and $$vv' \in E(H),$$vvźźE(H), or $$(2)\;v = v'$$(2)v=vź and $$uu' \in E(G)$$uuźźE(G). In this paper, for $$G = G_1 \times G_2$$G=G1×G2, where $$G_1$$G1 and $$G_2$$G2 are Hamiltonian graphs, IHC($$G_1 \times G_2) \ge $$G1×G2)ź IHC($$G_1)$$G1) or IHC($$G_1)$$G1) + 2 is proved when given some different conditions.