The Blakers-Massey element in $\pi_6(S^3)\cong\mathbb{Z}_{12}$ can be represented by such a map. This is done explicitly on page 3 of the paper https://arxiv.org/abs/math/0501091, published as 

<cite authors="Abresch, U.; Durán, C.; Püttmann, T.; Rigas, A.">_Abresch, U.; Durán, C.; Püttmann, T.; Rigas, A._, [**Wiedersehen metrics and exotic involutions of Euclidean spheres**](http://dx.doi.org/10.1515/CRELLE.2007.025), J. Reine Angew. Math. 605, 1-21 (2007). [ZBL1125.57017](https://zbmath.org/?q=an:1125.57017).</cite> 

Let $\mathbb{H}$ denote the quaternions, and represent the $6$-sphere as
$$
S^6 = \{(p,w)\in \mathbb{H}\times\mathbb{H} \mid \mathfrak{Re}(p)=0\mbox{ and } |p|^2+|w|^2=1\}.
$$
The map $b:S^6\to S^3\subseteq \mathbb{H}$ is given by 
$$
b(p,w) = \left\{\begin{array}{ll} \frac{w}{|w|} e^{\pi p} \frac{\overline w}{|w|}, & w\neq 0 \\
-1, & w=0, \end{array}\right.  
$$
where $e^{\pi p} = \cos(\pi |p|) + \sin(\pi|p|) \dfrac{p}{|p|}$ is the quaternionic exponential. 

The fact that $b(-p,-w)=\overline{b(p,w)}$ is easily checked (and is noted in the proof of Theorem 1 in the linked paper).