If your emission end time is not 1 with your start time it retains previous knowledge for the previous particles not yet re-emitted.
Setting the end to 1 and back again seems to reset it.
As I recall it was a painful thing in Houdini to go back to free 1 since it I guess in Blender terms cleared your bake.
Maybe if you animate it from frame zero you get the proper reset of all particles. Totally painful bug I'm sure will get fixed. This is happening on 2.79...
Oh wait this is not keyable either?! Lame... again. F! ... so you have to remember to do it manually before you render, or every time you want to start at the beginning and not see previous pre-emission positions I guess.
This might have something to do with one particle system getting keys from another particle system in 'Keyed' mode or my particulars. Alas I doubt that has anything to do with.
Update: It's also seemingly intermittently solving this for me. A tangle of trouble. Also I'm seeing a cache tab under emission now. I was trying to clean the Scene Cache and after completely deleting the Rigid Body World which I never baked did nothing.
Anyway it's glitched when it loops so it's definitely buggy. Maybe baking is the only way. No the baking returns with same problem particles from middle of animation on the first frame, not properly reset to their initial state. I know seems like a no-brainer right? Well somebodies pulling a no-brainer here and it's not me.