For a closed, connected direct product Riemannian manifold (M,g)=(M₁×⋯×M_l,g₁⊕⋯⊕g_l), we define its multiconformal class [[g]] as the totality {f₁²g₁⊕⋯⊕f_l²g_l} of all Riemannian metrics obtained from multiplying the metric g_i of each factor M_i by a function f_i²>0 on the total space M. A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g~] of every g~∈[[g]]. In this article, we prove that [[g]] carries a metric of positive scalar curvature if and only if the conformal class of some factor (M_i,g_i) does, under the technical assumption dimM_i≥2. We also show that, even in the case where every factor (M_i,g_i) has positive scalar curvature, [[g]] carries a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.