It's very possible some varieties are very prone to producing sports. You could take a cutting of the sported branch or try air layering. http://monsantiano.blogspot.com/ I have some friends who actually strip a small portion of the branches outer layer, (just like air layering) then peg it into the soil.
Do you have any pictures?

Thanks for the really neat link. I didn't know anything about that method. I did post this question on another forum.

Someone pointed out what I was asking about is a common occurrence with New Year rose. It is a new rose to me so I was unaware two entirely different looking blooms could be on the same bush. I learn something new every day. These are the two blooms growing on the New Year bush. A whole bunch of orange blooms and the one unusual one. What's even funnier, the orange blooms have a very mild fragrance and the unusual bloom has a very strong sweet fragrance.

Yeah the second rose just looks like an aging bloom that may be responding to fluctuating weather.