We consider the problem of certifying binary observables based on a Bell inequality violation alone, a task
known as self-testing of measurements. We introduce a family of commutation-based measures, which encode
all the distinct arrangements of two projective observables on a qubit. These quantities by construction take
into account the usual limitations of self-testing and since they are “weighted” by the (reduced) state, they
automatically deal with rank-deficient reduced density matrices. We show that these measures can be estimated
from the observed Bell violation in several scenarios and the proofs rely only on standard linear algebra. The
trade-offs turn out to be tight, and in particular, they give nontrivial statements for arbitrarily small violations. On
the other extreme, observing the maximal violation allows us to deduce precisely the form of the observables,
which immediately leads to a complete rigidity statement. In particular, we show that for all n 3 the n-partite
Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the n-partite Greenberger-Horne-Zeilinger state and
maximally incompatible qubit measurements on every party. Our results imply that any pair of projective
observables on a qubit can be certified in a truly robust manner. Finally, we show that commutation-based
measures give a convenient way of expressing relations among more than two observables.