  We study the multicritical behavior arising from the competition of two
distinct types of ordering characterized by O(n) symmetries. For this purpose,
we consider the renormalization-group flow for the most general $O(n_1)\oplus
O(n_2)$-symmetric Landau-Ginzburg-Wilson Hamiltonian involving two fields
$\phi_1$ and $\phi_2$ with $n_1$ and $n_2$ components respectively. In
particular, we determine in which cases, approaching the multicritical point,
one may observe the asymptotic enlargement of the symmetry to O(N) with
N=n_1+n_2. By performing a five-loop $\epsilon$-expansion computation we
determine the fixed points and their stability. It turns out that for
N=n_1+n_2\ge 3 the O(N)-symmetric fixed point is unstable. For N=3, the
multicritical behavior is described by the biconal fixed point with critical
exponents that are very close to the Heisenberg ones. For N\ge 4 and any
n_1,n_2 the critical behavior is controlled by the tetracritical decoupled
fixed point. We discuss the relevance of these results for some physically
interesting systems, in particular for anisotropic antiferromagnets in the
presence of a magnetic field and for high-T_c superconductors. Concerning the
SO(5) theory of superconductivity, we show that the bicritical O(5) fixed point
is unstable with a significant crossover exponent, $\phi_{4,4}\approx 0.15$;
this implies that the O(5) symmetry is not effectively realized at the point
where the antiferromagnetic and superconducting transition lines meet. The
multicritical behavior is either governed by the tetracritical decoupled fixed
point or is of first-order type if the system is outside its attraction domain.
