Magic Wizard Problem

(a) 10x10 magic square formed of two orthogonal latin squares. Found by Ernest Tilden Parker 1960. (See Frontispiece of Rouse Ball's Mathematical Recreations 12th and later editions)
(b) Permuted ranks and files so diagonal is 00, 01, ..., 08, 09.

(1) Find a permute that has the minimum number of different move types
between successive cells (00-01, 01-02, ..., 98-99).

Is a Magic Wizard tour possible? (No moves crossing an intermediate cell)
The 49-50 and 79-80 moves would have to be wazir steps {0,1}. Moves like {2,2}, {2,4}, {2,6}, {3,3}, {3,6} crossing other cell centres would be avoided. Maybe keep to moves of type {1,n} and {n,n+1}? i.e. just "off" being lateral or diagonal.

In the first try below the following 13 moves fail:16-17, 21-22, 33-34, 35-36, 51-52, 53-54, 64-65, 67-68, 88-89, 91-92, 93-94, 97-98, 99-00.

Is a Magic Queen tour possible? (all moves lateral or diagonal) Probably not.
Is a Magic Witch tour possible? (all moves crossing an intermediate cell)