  We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are
self-affine compact sets defined by two contraction affine mappings with the
same linear part. They are widely studied in the literature under various
names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications
in approximation theory, in the construction of multivariate Haar systems and
other wavelet bases, in the discrete geometry, and in the number theory. We
obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$
and show that they are all homeomorphic but not diffeomorphic. In the general,
non-isotropic, case it is proved that a 2-attractor is uniquely defined, up to
an affine similarity, by the spectrum of the dilation matrix. We estimate the
number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary
expanding polynomials with the free coefficient $\pm 2$. The total number of
such polynomials is estimated by the Mahler measure. We present several
infinite series of such polynomials. For some of the 2-attractors, their
H\"older exponents are found. Some of our results are extended to attractors
with an arbitrary number of digits.
