We develop a device-independent framework for testing quantum channels. That
is, we falsify a hypothesis about a quantum channel based only on an observed
set of input-output correlations. Formally, the problem consists of
characterizing the set of input-output correlations compatible with any
arbitrary given quantum channel. For binary (i.e., two input symbols, two
output symbols) correlations, we show that extremal correlations are always
achieved by orthogonal encodings and measurements, irrespective of whether or
not the channel preserves commutativity. We further provide a full, closed-form
characterization of the sets of binary correlations in the case of: i) any
dihedrally-covariant qubit channel (such as any Pauli and amplitude-damping
channels), and ii) any universally-covariant commutativity-preserving channel
in an arbitrary dimension (such as any erasure, depolarizing, universal
cloning, and universal transposition channels).