Can my crown be safely removed and put in the right way? Who should pay for this to be done right?

Hello, it's unfortunate that the crown is not comfortable in your mouth.  My guess would be that the contours of the crown are not matching that of your surrounding teeth which definitely would make it feel out of place.  The best thing would be go back to your dentist and see if they can adjust it until it feels better.  A worst case scenario would be to remove it and remake another one that feels more comfortable.  Good luck!

The chances of a crown fitting down all the way to the margins of the tooth preparation backwards is virtually impossible.  If somehow they ground the crown down in a way that they cemented it incorrectly then they would bare the cost of replacement of the crown.  It is okay for you to expect your crown to be placed and cemented in its correct orientation.  Did your dentist cement it in 180 degrees off or the assistant?  If the assistant performed the final cementing, I would suggest that you request the dentist to perform the work the next time.