They pretend to be brother and sister. They also pretend to be related to Esme so they live with the cullens.
Before they were turned into vampires Emmett was born Emmett McCarty and Jasper was born Jasper Whitlock. They did not have the same last name.
Jasper's last name was originally Whitlock but he changed it when he joined the Cullens with Alice. Jasper uses whatever last name matches the current cover story being used by the Cullens. In Forks, he and Rosalie are pretending to be twins and foster children of the Cullens, so it makes sense that their last name would be the same and not Cullen.
The members of the Cullen "family" are trying to keep up the pretense that they are actually related. Edward, Alice, and Emmett took on the name Cullen when they were "adopted" by Carlisle and Esme; but it would be creepy for Alice and Jasper (and Rosalie and Emmett) to have the same last name, since they're together romantically. So Rosalie kept her original last name and they pretend that Jasper is her twin brother, hence his last name.
In the Twilight series, Rosalie and Jasper have the same last name, Hale, because they were both turned into vampires by the same person, Carlisle Cullen. As a way to blend in, Carlisle often took in vampires with the last name "Cullen" to give the appearance of being a family.
The same reason Alice, Jasper, Emmett, Esme and the Olympic Clan have two last names human last name and vampire last name, e.g. Alice: Mary Alice Brandon Cullen (Brandon is her human last name, Mary is her Human name) Jasper: Jasper Whitlock Hale (Whitlock is his human last name) Emmett: Emmett McCarty Cullen (McCarty is his human last name) Esme: Esme Anne Platt Evenson Cullen (Platt and Evenson are her human last names) e.c.t. Renesmee is half human so she still gets her human last name as well as her vampire last name.
This is just a theory but maybe they have the same last nameI am nearing the end of the third book, and if Jasper and Rosalie have a closer relationship than adoptive brother and sister, it has not been explained yet. They have the same last name of Hale because they are posing as twins. Hale is Rosalie's real name from before she was turned in the 1930's. They could not be actual bother and sister, let alone twins, because in the third book Jasper is revealed to be much older than Rosalie. He was turned in the 1860's, and his real last name is Whitlock.
Jasper Didn't Jasper and Alice join at the same time? Alice had a vision and went to look and wait for Jasper, when she did they both went to the Cullens.
In the Twilight series, some of the siblings have different last names because they were adopted into the same family. For example, Rosalie Hale and Jasper Hale were adopted by the Cullen family, hence having different last names.
Jasper and Rosalie both have the last name Hale because they were both turned into vampires by the same person, Carlisle Cullen, and were then adopted into the Cullen family. The family chose to maintain the same last name for consistency and to help them blend in as a family unit.
Jasper County
Because they are all sapost to be adopted kids not look like their married when their 17-19 years old.
Emmett's last name is Cullen. Jasper's last name is Hale. You never find out what Emmett's real last name is, so the Cullen's just decided to make it Cullen. They wanted to stay kind of hidden and conspicuous, so they had Jasper share a last name with Rosalie. Jasper's original last name is Whitlock. Rosalie and Jasper are both pale, tall and extraordinarily beautiful, so they could get away with being twins.