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S-Heun operators on linear and $q$-linear grids are introduced. Theseoperators are special cases of Heun operators and are related to Sklyanin-likealgebras. The Continuous Hahn and Big $q$-Jacobi polynomials are functions onwhich these S-Heun operators have natural actions. We show that the S-Heunoperators encompass both the bispectral operators and Kalnins and Miller'sstructure operators. These four structure operators realize special limit casesof the trigonometric degeneration of the original Sklyanin algebra.Finite-dimensional representations of these algebras are obtained from atruncation condition. The corresponding representation bases are finitefamilies of polynomials: the para-Krawtchouk and $q$-para-Krawtchouk ones. Anatural algebraic interpretation of these polynomials that had been missing isthus obtained. We also recover the Heun operators attached to the correspondingbispectral problems as quadratic combinations of the S-Heun operators

Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials