It sure looks like a rose and from the looks of it, it has been there for awhile. It is probably one of the old small flowered roses.. can you get us a picture when it is blooming, which should be sooner..

Thank you lelamsk for your reply. Here is a photo of the currently blooming flowers ... Looking forward to figuring this one out. Thanks!

It looks like the species (wild) rose called Cherokee Rose.

I checked the old rose section in my huge rose identification book, and the closest rose with 4 pure white petal and that wonderful yellow center was an Rosa brunonii "The Himalayan Musk".. seems very common in California. you didn't say where you were from.. but you have a beautiful old rose. Lela

Thank you for your replies ... Unfortunately it's much too big for the space it's in. I live in CT. Any thoughts in transplanting or cutting back? Thanks!

It also looks like they are invasive in the USA ... The Cherokee rose ... Which. Makes sense why it's growing like a beast! May just remove it all together. Thanks for your help seil

Most of the flowers I see in those photos have 5 petals, not 4.

There is a wild (species) invasive rose (with great fragrance!) that grows all over Southern New England, and this looks just like it...

Definitely a rose. I think it's probably root stock, though it's so pretty, perhaps not. It sounds so hardy that I suppose transplanting or cutting back hard would do fine. Once you cut it back a lot, it would be much easier to maintain.