From this page you can download Latin squares of orders up to 17 that have any specified autoparatopism. At least, you can specify the cycle structure of the autoparatopism, and from this it is easy to get any autoparatopism you might want. Note that, autoparatopisms that happen to be autotopisms are handled on another page so I won't give them here.
This work was performed for our paper:
M. J. L. Mendis and I. M. Wanless, Autoparatopisms of quasigroups and Latin squares.
We first give those Latin squares that have an autoparatopism of the form (ε,ε,γ;(12)). These were constructed in Theorem 4.7 of our paper and the examples offered here are precisely as given in the paper.
For all other autoparatopisms of the form (ε,β,γ;(12)), we now give an example that was found using a simple backtracking search (first filling in a subsquare on the rows and columns indexed by the fixed points of β). In this case we assume that β and γ are canonical permutations (which isn't the case in the examples above). The examples are given in different files according to the order of the Latin squares. Within each file the squares are ordered lexicographically by the cycle structures of the pair (β,γ), which exactly matches the order that autoparatopisms are listed in the paper.
Next we give the equivalent lists for all autoparatopisms of the form (ε,ε,γ;(123)). Again we assume that γ is a canonical permutation and order the squares lexicographically by the cycle structure of γ, which matches the order given in the paper. These examples were found by a similar search to those above, with a totally symmetric subsquare first being installed on the rows and columns indexed by the fixed points of γ.
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