It's not uncommon for rose plants to be grafted onto hardier root stocks. This helps the rose survive in colder climates, but it can also lead to a problem with suckers.
What is a Rose Sucker?Suckers are stalks that emerge from below the bud union, where the rose bush was grafted onto the root stock. Suckers are growing from the root stock and will not bloom, like the top half of your rose bush. However they will sap energy from the plant and can take over completely, so it's best to remove them. The only sure sign that a branch is a sucker and not simply new growth is that it is coming from below the bud union, although most suckers will have leaves that don't exactly resemble those on the upper portion of the rose bush.
How to Remove Rose SuckersCutting suckers with pruners seems to encourage more suckers. It's recommended that you dig down to where the sucker is originating and pull, twist or tear it off. Wear thick, protective gloves. You may have to follow the sucker back to its source. Suckers can pop up several feet from the originating rose bush.
The good news is that many newer roses are grown from cuttings, not grafts, and these do not sucker.