We study the symmetricity of the Whitehead element w_n ∊ π_<2np-3> (S^<2n-1>) for an odd prime p. It is shown that w_n considered as a map S^<2np-3> → S^<2n-1> factors through the p-fold covering map σ : S^<2np-3> → L^<2np-3> only when n is a power of p, and that w_<p^i>, actually factors through σ if 0 ≤ i ≤ 4. This is some of an odd prime version of the results of Randall and Lin for the projectivity of the Whitehead product [l_<2n-1>, l_<2n-1>] ∊ π_<4n-3>(S^<2n-1>).