  We study the so-called sign involutions on twisted forms of abelian
varieties, and show that such a sign involution exists if and only if the class
in the Weil--Ch\^{a}telet group is annihilated by two. If these equivalent
conditions hold, we prove that the Picard scheme of the quotient is \'etale and
contains no points of finite order. In dimension one, such quotients are
Brauer--Severi curves, and we analyze the ensuing embeddings of the genus-one
curve into twisted forms of Hirzebruch surfaces and weighted projective spaces.
