  We give a new characterization of biinfinite Sturmian sequences in terms of
indistinguishable asymptotic pairs. Two asymptotic sequences on a full
$\mathbb{Z}$-shift are indistinguishable if the sets of occurrences of every
pattern in each sequence coincide up to a finitely supported permutation. This
characterization can be seen as an extension to biinfinite sequences of
Pirillo's theorem which characterizes Christoffel words. Furthermore, we
provide a full characterization of indistinguishable asymptotic pairs on
arbitrary alphabets using substitutions and biinfinite characteristic Sturmian
sequences. The proof is based on the well-known notion of derived sequences.
