  Given an algebra with an idempotent, we introduce two procedures to construct
families of new algebras, termed mirror-reflective algebras and reduced
mirror-reflective algebras. We then establish connections among these algebras
by recollements of derived module categories. In case of given algebras being
gendo-symmetric, we show that the (reduced) mirror-reflective algebras are
symmetric and provide new methods to construct systematically both higher
dimensional (minimal) Auslander-Gorenstein algebras and gendo-symmetric
algebras of higher dominant dimensions. This leads to a new formulation of
Tachikawa's second conjecture for symmetric algebras in terms of idempotent
stratifications.
