  We perform a comprehensive study of the signatures of Lorentz violation in
electrodynamics on the Cosmic Microwave Background (CMB) anisotropies. In the
framework of the minimal Standard Model Extension (SME), we consider effects
generated by renormalizable operators, both CPT-odd and CPT-even. These
operators are responsible for sourcing, respectively, cosmic birefringence and
circular polarization. We propagate jointly the effects of all the relevant
Lorentz-violating parameters to CMB observables and provide constraints with
the most recent CMB datasets. We bound the CPT-even coefficient to $k_{F,E+B} <
2.31 \times 10^{-31}$ at 95\% CL. This improves previous CMB bounds by one
order of magnitude. The limits we obtain on the CPT-odd coefficients, i.e.
$|k_{(V)00}^{(3)}| < 1.54 \times 10^{-44} \; {\rm GeV}$ and $|\mathbf{k_{AF}}|
< 0.74 \times 10^{-44} \; {\rm GeV}$ at 95\% CL, are respectively one and two
orders of magnitude stronger than previous CMB-based limits, superseding also
bounds from non-CMB searches. This analysis provides the strongest constraints
to date on CPT-violating coefficients in the minimal SME from CMB searches.
