What is the difference between the Accepted reading and the Satoshis reading? They never match. Why?

Quick explanation - during mining partial solutions are found to a currency's proof-of-work algorithm. Until a partial solution is found (i.e. an accepted submission), a miner doesn't have anything that will validate. Because work items change with every new transaction on the network, when a miner stops, whatever work they get when they reconnect is almost always different, so progress is effectively reset since no submission was made. The pool also has no idea whether a miner is going to reconnect, so a miner's "slot" can be reassigned to another miner, as these are assigned from the pool of available slots (literally a numeric range in the proof-of-work) at the time a miner connects.

I have also been watching the counter for the Satoshis Mined, the one with the 3 digits aft the decimal point and is the main attraction of the app besides the ads. What I have noticed is that it goes backwards, i.e. counts back down, or seems have lost its place and goes down a few Satoshis, etc. and yet the 'rejected' pile didn't increase. Should not the Satoshis Mined just increase only? Are all of the 'accepted' being applied as mined, thus accrued, thus payable? What the app appears to be doing is losing Satoshis Mined ....

We'd definitely like to improve the integration between the pool site and the app - right now the app only reports the amounts mined during a session rather than all time.

The main thing would be showing what numbers users are interested in - i.e. mined in the session vs. mined in all sessions but pending vs. mined in all sessions and pending and paid out.