  We study which algebras have tilting modules that are both generated and
cogenerated by projective-injective modules. Crawley-Boevey and Sauter have
shown that Auslander algebras have such tilting modules; and for algebras of
global dimension $2$, Auslander algebras are classified by the existence of
such tilting modules.
  In this paper, we show that the existence of such a tilting module is
equivalent to the algebra having dominant dimension at least $2$, independent
of its global dimension. In general such a tilting module is not necessarily
cotilting. Here, we show that the algebras which have a tilting-cotilting
module generated-cogenerated by projective-injective modules are precisely
$1$-Auslander-Gorenstein algebras.
  When considering such a tilting module, without the assumption that it is
cotilting, we study the global dimension of its endomorphism algebra, and
discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as
special cases, we show that triangular matrix algebras obtained from Auslander
algebras and certain injective modules, have such a tilting module. We also
give a description of which Nakayama algebras have such a tilting module.
