So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface $$\Sigma $$ Σ in $$\mathbf{R}^{2n}$$ R 2 n are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface $$\Sigma $$ Σ in $$\mathbf{R}^{2n}$$ R 2 n when there exist only finitely many geometrically distinct symmetric closed characteristics on $$\Sigma $$ Σ , which extend the identity established by Liu and Long (J Differ Equ 255:2952–2980, 2013) of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu et al. (J Funct Anal 166:5598–5638, 2014) for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in $$\mathbf{R}^4$$ R 4 , then both of them must be elliptic.