A new grid conversion method is proposed to resample between two 2-D periodic lattices with the same sampling density. The main feature of our approach is the symmetric reversibility, which means that when using the same algorithm for the converse operation, then the initial data is recovered exactly. To that purpose, we decompose the lattice conversion process into (at most) three successive shear operations. The translations along the shear directions are implemented by 1-D fractional delay operators, which revert to simple 1-D convolutions, with appropriate filters that yield the property of symmetric reversibility. We show that the method is fast and provides high-quality resampled images. Applications of our approach can be found in various settings, such as grid conversion between the hexagonal and the Cartesian lattice, or fast implementation of affine transformations such as rotations.