  We study IE-closed subcategories of a module category, subcategories which
are closed under taking Images and Extensions. We investigate the relation
between IE-closed subcategories and torsion pairs, and characterize
$\tau$-tilting finite algebras using IE-closed subcategories. For the
hereditary case, we show that IE-closed subcategories can be classified by twin
rigid modules, pairs of rigid modules satisfying some homological conditions.
Moreover, we introduce mutation of twin rigid modules analogously to tilting
modules, which gives a way to calculate all twin rigid modules for the
representation-finite case.
