Check the oil for metal particles,a sure sign the cam lobes are being worn away by the tappets.
I wouldn't say you could fit the tappets upside down,they would physically fit but would look wrong.
There's a bedding in procedure if you have a new cam/tappets,possibly if you have been trying to start it with no success it has worn a lobe down which causes the valves not yo open-cue tapping and running like a bag of spanners.
Pull the rocker cover off and turn it over by hand observing the lift and duration of each rocker,if there's 1 not moving much there's your worn lobe.