Note: This may be related to MC-10174, but it seemed sufficiently different to me to warrant a post (this one is control-able, while the other is random).

Blocks in the starting dropper are moving different numbers of droppers based on whether the power source is hand-placed or toggled.

With the provided setup, when a redstone torch is hand-placed under the dropper, it moves along two droppers (ends up kitty-corner to where it started). If the redstone torch is toggled (with a lever, line of redstone with another torch, etc.) the block only moves one dropper (ends up in the dropper to the right).

Setup: See attached picture. 1 furnace, 5 droppers. The dropper under the furnace is pointing to the right. The dropper to the right of that one is pointing up, then another pointing up, then pointing left, then pointing down into the furnace (a loop into the top of the furnace).

Place an item in the dropper under the furnace and place a redstone torch under that dropper. Different results will be obtained of the torch is hand-placed versus toggled with the lever.