A hierarchy of clopen graphs on the Baire space

We say that E ⊆ X × X is a clopen graph on X iff E is symmetric and irreflexive and clopen relative to X\∆ where ∆ = {(x, x) : x ∈ X} is the diagonal. Equivalently E ⊆ [X] and for all x 6= y ∈ X there are open neighborhoods x ∈ U and y ∈ V such that either U × V ⊆ E or U × V ⊆ X\E. For clopen graphs E1, E2 on spaces X1, X2, we say that E1 continuously reduces to E2 iff there is a continuous map f : X1 → X2 such that for every x, y ∈ X1 (x, y) ∈ E1 iff (f(x), f(y)) ∈ E2.