What is happening is that when you stop a timeline all of it's state is discarded and the instances are left at their ending position. If you then start playing it again all of it's values are applied to the corresponding instances relative to their new transformation which is different to the initial transformation in the editor.
In this case the initial editor state is the spider with the legs in a flat position. That is why the timeline works the first time, all of it's values are applied in relation to that initial transformation, when you play it again though, that is not the case any more so it starts looking strange.
What you can do in this particular case is pausing and resuming the timeline, rather than stopping it and playing it again. When you pause it, the state is not thrown away altogether, so resuming it works as you would expect.
Will likely add another action to give the option to reset a hierarchy in a timeline to it's initial pose.