  Type Ia supernovae (SNe) are believed to be caused by the thermonuclear
explosion of a white dwarf (WD), but the nature of the progenitor system(s) is
still unclear. Recent theoretical and observational developments have led to
renewed interest in double degenerate models, in particular the "helium-ignited
violent merger" or "dynamically-driven double-degenerate double-detonation"
(D$^6$). In this paper we take the output of an existing D$^6$ SN model and
carry it into the supernova remnant (SNR) phase up to 4000 years after the
explosion, past the time when all the ejecta have been shocked. Assuming a
uniform ambient medium, we reveal specific signatures of the explosion
mechanism and spatial variations intrinsic to the ejecta. The first detonation
produces an ejecta tail visible at early times, while the second detonation
leaves a central density peak in the ejecta that is visible at late times. The
SNR shell is off-centre at all times, because of an initial velocity shift due
to binary motion. The companion WD produces a large conical shadow in the
ejecta, visible in projection as a dark patch surrounded by a bright ring. This
is a clear and long-lasting feature that is localized, and its impact on the
observed morphology is dependent on the viewing angle of the SNR. These results
offer a new way to diagnose the explosion mechanism and progenitor system using
observations of a Type Ia SNR.
