Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma, $$ |V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}. $$ Since $g-I$ and its dual map $g^*-I$ have kernels of the same dimension, it follows that $|V/G|=|V^*/G|$.

The above argument shows that a vector space and its dual have the same number of orbits under the action of linear group. Can it happen that the cardinalities of the orbits are different?