The dominant dimension of algebras in the class $$\mathbf A $$ A of 1-quasi-hereditary algebras introduced in Pučinskaitė (J Lond Math Soc (2) 87(2):478–496, 2013) is at least two. By the Morita–Tachikawa Theorem this implies that $$\mathbf A $$ A is related to a certain class $$\mathbf B $$ B of pairs consisting of an (another) algebra and a module satisfying the double centralizer condition. In this paper we determine the class $$\mathbf B $$ B , and show the connection between the partial order of a 1-quasi-hereditary algebra and the structure of the related pair in $$\mathbf B $$ B (see Theorem A). If the first component of a pair in $$\mathbf B $$ B is a commutative algebra, then the corresponding algebra in $$\mathbf A $$ A receives additional features described in Theorem B. Finally we introduce the Ringel dual for objects in $$\mathbf B $$ B . Of particular interest are those pairs in $$\mathbf B $$ B which occur as Ringel dual (see Theorem C).