  For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold
with boundary $(\bar{X},\bar{g})$, we give a natural construction of the
Calder\'on projector and of the associated Bergman projector on the space of
harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our
approach is based on the conformal covariance of $D_{\bar{g}}$ and the
scattering theory for the Dirac operator associated to the complete conformal
metric $g=\bar{g}/\rho^2$ where $\rho$ is a smooth function on $\bar{X}$ which
equals the distance to the boundary near $\partial\bar{X}$. We show that $({\rm
Id}+\tilde{S}(0))/2$ is the orthogonal Calder\'on projector, where
$\tilde{S}(\lambda)$ is the holomorphic family in $\{\Re(\lambda)\geq 0\}$ of
normalized scattering operators constructed in our previous work, which are
classical pseudo-differential of order $2\lambda$. Finally we construct natural
conformally covariant odd powers of the Dirac operator on any spin manifold.
