In this study, we succeeded in synthesizing new antiperovskite phosphides $M$Pd$_3$P ($M$ = Ca, Sr, Ba) and discovered the appearance of a superconducting phase (0.17 $leq$ $x$ $leq$ 0.55) in a solid solution (Ca$_{1-x}$Sr$_x$)Pd$_3$P. Three perovskite-related crystal structures were identified in (Ca$_{1-x}$Sr$_x$)Pd$_3$P and a phase diagram was built on the basis of experimental results. The first phase transition from centrosymmetric ($Pnma$) to non-centrosymmetric orthorhombic ($Aba$2) occurred in CaPd$_3$P near room temperature. The phase transition temperature decreased as Ca$^{2+}$ was replaced with a larger-sized isovalent Sr$^{2+}$. Bulk superconductivity at a critical temperature ($T$$_c$) of approximately 3.5 K was observed in a range of $x$ = 0.17 - 0.55; this was associated with the centrosymmetric orthorhombic phase. Thereafter, a non-centrosymmetric tetragonal phase ($I$41$md$) remained stable for 0.6 $leq$ $x$ $leq$ 1.0, and superconductivity was significantly suppressed as samples with $x$ = 0.75 and 1.0 showed ($T$$_c$) values as low as 0.32 K and 57 mK, respectively. For further substitution with a larger-sized isovalent Ba$^{2+}$, namely (Sr$_{1-y}$Ba$_y$)Pd$_3$P, the tetragonal phase continued throughout the composition range. BaPd$_3$P no longer showed superconductivity down to 20 mK. Since the inversion symmetry of structure and superconductivity can be precisely controlled in (Ca$_{1-x}$Sr$_x$)Pd$_3$P, this material may offer a unique opportunity to study the relationship between inversion symmetry and superconductivity.
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