The δ expansion method is used to study all scalar field theories with symmetric monomial self-interactions of the form (φ2)p in four dimensions. With a momentum cut-off Λ it is shown that these theories are all distinct, admit spontaneous symmetry breaking, and have a nonperturbative structure in the coupling constant λ. In the limit that the momentum cut-off is taken to infinity, it is shown that the δ expansion of all these theories degenerates to the renormalized expansion of a single theory independent of p. That expansion is the renormalized weak-coupling expansion of λφ4. It is argued that all theories of this class are the same in the limit Λ → ∞. This universal behavior of all monomial interactions with the discrete symmetry φ → −φ is interpreted as a strong indication that all such theories are free, as λφ4 is believed to be.