The Salsa20 core preserves diagonal shifts? What do we mean by 


  the Salsa core preserves diagonal shifts
The input of the Salsa20 core is a 4x4 array of 32-bit words. If you move all the inputs diagonally (wrapping), the output will be the same as the original output, except that it's moved by the same offset.

$P^\prime_{i,j}=P_{i+k,j+k}$ implies $C^\prime_{i,j}=C_{i+k,j+k}$ (indices reduced modulo 4)

This symmetry follows directly from Salsa20 applying the same operations to all columns and all rows.

The full Salsa20 uses four fixed input words, which ensures that no matter how you choose the remaining inputs, you can never find two different inputs which could be transformed into each other using such a diagonal shift.