Mint reproduces two different ways. It sends up new plants from the roots. It sends a lot of new plants up by the roots. That is part of why it is considered so invasive. The plants that come up from the roots are clones and will taste exactly like the parent plant.
Mint reproduces by seeds. Mint produces a whole lot of seeds. These seeds are pretty hardy. I don't know if they are spread by animals, lke birds eating the seeds, but they will drop to the ground around the parent plant and sprout. Not every mint plant coming up around the parent plant is from the roots.
The flavor of the plants coming up from seeds will depend on the parent plants that contributed the genetics, both the original seed from the parent plant and whichever mint plant contributed the pollen. I don't know if mint is like apples and requires another variety to pollinate its seeds or if it can pollinate itself with a little help from pollinators. If it self-pollinates, I don't know if it always breeds true or ir it is like potatoes and can vary in flavor even if it self pollinates. I strongly suspect it would breed true since they sell seeds for specific varieties, but there is a lot I don't know for sure.
When mint flowers, you will notice a lot of bees and other pollinators on it. Some bees can travel a mile or more looking for nectar. Just moving your mint around to the other side of your house will not absolutely guarantee that the mint will not be cross pollinated with a different variety. I strongly suspect that mint can be pollinated by its own pollen and that it breeds true. If that is the case, then the further away it is from other varieties, then the less likely it is the pollenators will bring pollen from other varieties, so the seeds are more likely to breed true.
Some people feel that the parent plant sends so many sprouts up from the roots that any seeds that drop have a hard time sprouting and developing. I suspect the closer you are to the center of the original plant, there might be some validity in that, but the further away you go from the center, the less of a guarantee there is. Again, something I really don't know.
I will say I did plant three different varieties in widely separated locations, one on the opposite side of the house, two years ago and I have not noticed any mixing yet. But I don't consider two years long enough to give any real assurances.
I don't know if this helps or not. Personally I would transplant it.
Good luck!
Herbgardener 
