  Hypersymplectic structures with torsion on Lie algebroids are investigated.
We show that each hypersymplectic structure with torsion on a Lie algebroid
determines three Nijenhuis morphisms. From a contravariant point of view, these
structures are twisted Poisson structures. We prove the existence of a
one-to-one correspondence between hypersymplectic structures with torsion and
hyperk\"{a}hler structures with torsion. We show that given a Lie algebroid
with a hypersymplectic structure with torsion, the deformation of the Lie
algebroid structure by any of the transition morphisms does not affect the
hypersymplectic structure with torsion. We also show that if a triplet of
$2$-forms is a hypersymplectic structure with torsion on a Lie algebroid $A$,
then the triplet of the inverse bivectors is a hypersymplectic structure with
torsion for a certain Lie algebroid structure on the dual $A^*$, and
conversely. Examples of hypersymplectic structures with torsion are included.
