The present paper proposes that reconstruction scheme and interpolation scheme can be converted into each other through two series of adapter schemes, which include reconstruction-to-interpolation (RI) adapter schemes and interpolation-to-reconstruction (IR) adapter schemes. For the high-order spatial discretization of the compressible Navier-Stokes equations, the RI adapter schemes can be used to derive interpolation schemes for the interpolation-based cell-centered finite difference method from the available optimized reconstruction schemes. The main advantage of the interpolation-based cell-centered finite difference method is the capability to realize high-order discretization on curvilinear grids with both shock-capturing capability and satisfaction of the geometric conservation law. In the present paper, we first derive the IR adapter schemes by comparing the difference schemes with their strong conservative forms. We then develop the corresponding RI adapter schemes by inversing the IR adapter schemes. Thereafter, the applications to the one-dimensional linear wave equation and the one-dimensional inviscid Burgers' equation have been briefly discussed. Finally, to demonstrate the application to three-dimensional Navier-Stokes equations, three highly optimized nonlinear reconstruction schemes are adapted into the corresponding interpolation ones through RI adapter schemes, which include WENO-CU6, WGVC-WEN07 and OMP6 schemes. The new interpolation schemes from adapters are compared with their original reconstruction ones through several benchmark cases. No noticeable robustness loss or accuracy loss has been found in these cases, indicating the effectiveness of the adapter schemes. No obvious increase in time cost has been observed, indicating the efficiency of the adapter schemes. (C) 2019 Elsevier Inc. All rights reserved.