In the vicinity of a tricritical point an n-component spin system is shown to have continuous transitions which are driven by fluctuations (they would be first order according to Landau's theory). We show that spin anisotropies which imply crossover to lower symmetry (e.g., of m-component spins with m<n) may turn these fluctuation-driven continuous transitions first order via tricritical points. In cubic systems, which exhibit fluctuation-driven first-order transitions, the anisotropy may yield two consecutive tricritical points. We present a detailed renormalization-group analysis of these situations with emphasis on the importance of the sixth-order terms in the Ginzburg-Landau-Wilson continuous-spin Hamiltonian. A list of possible experimental realizations is also given.