The phase diagrams of systems described by a Hamiltonian containing an anisotropic quadratic term of the form 12gα= 1ncαx→Sα2(x→), and a cubic anisotropic term να=1nx→Sα4(x→), are studied using mean-field theory, scaling theory, and expansions in ε(=4-d) and 1n. Here, Sα(x→) (a=1, ..., n) is a local n-component ordering variable. Systems to which the analysis is applicable include perovskite crystals, stressed along the  direction (n=3), anisotropic antiferromagnets in a uniform field, uniaxially anisotropic ferromagnets, ferroelectric ferromagnets and crystalline He4(n=2). When g=0 and T=Tc these systems undergo a phase transition that may be associated (for small n) with the Heisenberg fixed point (ν*=0) or (otherwise) with the cubic fixed point (ν*>0) of the renormalization group. Although ν is an "irrelevant variable" in the former case, it is found to have important effects. For ν<0, the point g=0, T=Tc represents a bicritical point in the g-T plane, at which a first-order "spin-flop" line (separating two distinct ordered phases) meets two critical lines. For ν>0, the "flop" line splits into two critical lines, associated with transitions between each of the ordered phases and a new intermediate phase; the point T=Tc, g=0 is then tetracritical. The shape of the boundary of the intermediate phase is given by T=T2(g, ν) with [Tc-T2(g, ν)]∼(gν)1ψ2, where ψ2=φg-φν (if the tetracritical point is Heisenberg-like) or ψ2=φgC (if it is cubic). Here, φg, φν, and φgC are appropriate crossover exponents associated with the two symmetry-breaking perturbations. The phase diagram of  -stressed perovskites is also discussed and the experimental situation briefly reviewed.