Here is a strongly invertible diagram of 10_49. The symmetry axis is almost horizontal.

This was created by loading the 'standard' diagram of 10_49 to KLO and then using KLO's simplifying method together with some work by hand. (KLO's simplifying method seems to prefer symmetric diagrams this is why I would expect the method to work in most cases.)

We have used similar methods for example here to create strongly invertible surgery diagrams. I would expect that the same works for the other 7 knots from your list. If you need help with that let me know.

EDIT: For 10_62 and 10_65 it works similar. Here are diagrams of them:

Here is a strongly invertible diagram of 10_49. The symmetry axis is almost horizontal.

This was created by loading the 'standard' diagram of 10_49 to KLO and then using KLO's simplifying method together with some work by hand. (KLO's simplifying method seems to prefer symmetric diagrams this is why I would expect the method to work in most cases.)

We have used similar methods for example here to create strongly invertible surgery diagrams. I would expect that the same works for the other 7 knots from your list. If you need help with that let me know.

EDIT: For the other knots it works similar. Here are diagrams of them.

10_62 10_65 10_113 10_143 10_152 10_154

(For the last two the symmetry axis is orthogonal to the projection plane.)

Here is a strongly invertible diagram of 10_49. The symmetry axis is almost horizontal.

This was created by loading the 'standard' diagram of 10_49 to KLO and then using KLO's simplifying method together with some work by hand. (KLO's simplifying method seems to prefer symmetric diagrams this is why I would expect the method to work in most cases.)

We have used similar methods for example here to create strongly invertible surgery diagrams. I would expect that the same works for the other 7 knots from your list. If you need help with that let me know.

Here is a strongly invertible diagram of 10_49. The symmetry axis is almost horizontal.

This was created by loading the 'standard' diagram of 10_49 to KLO and then using KLO's simplifying method together with some work by hand. (KLO's simplifying method seems to prefer symmetric diagrams this is why I would expect the method to work in most cases.)

We have used similar methods for example here to create strongly invertible surgery diagrams. I would expect that the same works for the other 7 knots from your list. If you need help with that let me know.

EDIT: For 10_62 and 10_65 it works similar. Here are diagrams of them: