The Clawshot is a special item in Twilight Princess. Unlike with other items, Link cannot move normally while the clawshot is traveling. With L-slides, though, it is possible to travel a small distance in any direction while the clawshot is moving. These slides can save time, as in Palace of Twilight while clawshotting sols, and can also be used to initiate Clawshot Actor Displacement.
L-sliding any direction other than forwards or backwards has a maximum 6 speed. L-sliding backwards is maximum 2.5 speed, and L-sliding forwards is maximum 1.9 speed. This means that it is almost always faster to L-slide diagonally than directly forwards or backwards even if you want to go directly forwards or backwards.
Alternatively:
In either case, the camera will return to third person and Link will slide until the clawshot returns to his hand. If you use the C-stick to turn the camera while L-sliding, do not release the C-stick to neutral until after letting go of L so that the camera doesn't snap back behind Link.