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In this paper, we study the boundedness of pseudodifferential operators withsymbols in the H\"ormander class $S^0_{\rho,\rho}$ on $\alpha$-modulationspaces $M_{p,q}^{s,\alpha}$, and consider the relation between $\alpha$ and$\rho$. In particular, we show that pseudodifferential operators with symbolsin $S^0_{\alpha,\alpha}$ are bounded on all $\alpha$-modulation spaces$M^{s,\alpha}_{p,q}$, for arbitrary $s\in\mathbb{R}$ and for the whole range ofexponents $0 < p,q \leq \infty$.

Pseudodifferential operators with symbols in the H\"ormander class $S^0_{\alpha,\alpha}$ on $\alpha$-modulation spaces