  We introduce the twisted $\boldsymbol{\mu}_4$-normal form for elliptic
curves, deriving in particular addition algorithms with complexity $9\mathbf{M}
+ 2\mathbf{S}$ and doubling algorithms with complexity $2\mathbf{M} +
5\mathbf{S} + 2\mathbf{m}$ over a binary field. Every ordinary elliptic curve
over a finite field of characteristic 2 is isomorphic to one in this family.
This improvement to the addition algorithm, applicable to a larger class of
curves, is comparable to the $7\mathbf{M} + 2\mathbf{S}$ achieved for the
$\boldsymbol{\mu}_4$-normal form, and replaces the previously best known
complexity of $13\mathbf{M} + 3\mathbf{S}$ on L\'opez-Dahab models applicable
to these twisted curves. The derived doubling algorithm is essentially optimal,
without any assumption of special cases. We show moreover that the Montgomery
scalar multiplication with point recovery carries over to the twisted models,
giving symmetric scalar multiplication adapted to protect against side channel
attacks, with a cost of $4\mathbf{M} + 4\mathbf{S} + 1\mathbf{m}_t +
2\mathbf{m}_c$ per bit. In characteristic different from 2, we establish a
linear isomorphism with the twisted Edwards model over the base field. This
work, in complement to the introduction of $\boldsymbol{\mu}_4$-normal form,
fills the lacuna in the body of work on efficient arithmetic on elliptic curves
over binary fields, explained by this common framework for elliptic curves in
$\boldsymbol{\mu}_4$-normal form over a field of any characteristic. The
improvements are analogous to those which the Edwards and twisted Edwards
models achieved for elliptic curves over finite fields of odd characteristic,
and extend $\boldsymbol{\mu}_4$-normal form to cover the binary NIST curves.
