Are some mechanical relays sensitive to their orientation?
Operating position is frequently specified on motor starter relay installation guides, so it can matter. For many of them it doesn't- they are specified as "non-positional". 

An educated guess as to the root cause-  crap from the contacts as they wear falling into the wrong places as they age. Which would mean that, after a period of heavy use, flipping it over for shipping or whatever could cause problems even if it was mounted properly. 







Reading the datasheet and application information such as installation guides is the best approach. 



Of course there are other reasons that a relay or contactor could stick closed or stay open, such as overloading, contact welding due to a short circuit, poor design of the relay or contact metallurgy etc. 

Hg-displacement relays will stay closed if you flip them upside down, because they are filled with liquid.