The closest I think we can come is 'Autumn Damask,' which will bloom heavily in the spring.
When mature, it can produce Fall bloom.
There is also a "Pickering Four Seasons Rose," which is something like a deep fuchsia version of Autumn Damask. More or less.

But these are not the species rose, R. damascena.
That rose is not remontant.

Jeri

As far as I know, there are no climbing damasks, and Autumn Damask, as Jeri implies, is not a great repeater.

Many hybrid tea roses have strong damask scent, for example Perfume Delight (bright pink) and the famous dark reds, but the latter may burn in the desert sun.

I can't think of a pink climber with damask scent.

I know you asked for Damask roses, but "Evelyn," a David Austin rose was selected by Crabtree & Evelyn to represent them.

Again, not a damask, but maybe Gertrude Jekyll?
Masha

Here is a link that might be useful: GT at Paul Barden's Roses

There you go, Masha. Gertrude Jekyll has powerful damask scent and nasty damask thorns. Canes can run as long as 8'. I kept it as a small climber for a while. It was terribly susceptible to blackspot (as are most damasks), but that won't be a concern in Arizona. The flower is superb and better than any damask. Repeat is not as good as the better modern climbers.

Here is a link that might be useful: GJ at HMF

Hi Arizsun, lots of good ideas here.

I have to say that the real rose in bloom is much better than anything Crabtree & Evelyn or any other perfume maker have managed to capture in a bottle. A few of the red hybrid perpetuals have somehow managed to inherit the damacenner type fragrance, and again have passed it down to a few hybrid teas. These hybrid teas have again sported climbers. I grow Climbing Etoille de Hollande which is one of them. Going for either of these climbing hybrid teas is not a bad idea or perhaps one or two of the hybrid perpetuals General Jacqueminot or Souvenir du Docteur Jamain.

If you ever have rom for a repeat growing shrub, Comte de Chambord and Rose de Resht are well worth planting.

Not climbing, but Rose de Resht is probably the rose with most damacena blood and still very good repeat bloom, and arguably most fragrance of them all.

OMG! (seriously :D)
You people are so knowledgeable.. Hours of my research wouldn't find those answers. Thank you so much.Not only that, helpful names and links...

I would have never imagined "tea rose" can smell like damascus. Thank you for teaching me that.

I mentioned crabapple and Evelyn because that's the closest I can find in the US to an Eau de perfume I bought in France long time ago - no name, no manufacture on the bottle.I have been looking for that scent and didn't know it was damascus scent until very recently.

I have been only growing edibles but I have a best spot for a rose. Thank you again. I will start my research and find the best one for me.

I would have never imagined "tea rose" can smell like damascus.

*** I don't believe there are any Tea Roses which have the Damask fragrance.

There are, however, quite a few HYBRID Tea Roses which have that fragrance.

Tea Roses are ancestors of the modern Hybrid Tea Roses, and are rarely available at local nurseries.

Hybrid Tea Roses are available at most local nurseries.

Jeri

Fragrance is different for everyone! Roses Like Chrysler Imperial Oklahoma, Mr Lincoln all have very strong fragrances that some claim to be very similar to Damask.
David Austin's Munstead wood also has a very strong "old rose fragrance"
The Portland Rembrandt has a very very beautiful damask like fragrance as does Rose De Rescht!

If you like damask scent arizsun, you might like to try something I bought at Target, Botanic's Rosewater Toning Spritz, 100% Organic. It was developed with plant extracts from Kew Royal Botanic Gardens. It smells heavenly!!! Very strong damask fragrance. Ingredients: Aqua (water), Alcohol denat, Glycerin, Rosa damascena flower oil, Citronellol.

I grow both Rose de Rescht and Pickering Four Seasons. Both have wonderful scent, but both are also smallish plants and don't climb. RdR has the better rebloom.

I have a magnificent damask with a powerful fragrance called York and Lancaster which I've grown for over 20 years in partial shade (bottom of plant in full shade, top of plant in full sun most of the day). It is now up on my roof but would only be able to climb with lots of support (wants to be floppy). However there is zero rebloom.

You might want to try Portland from Glendora. Wonderful, strong scent. This rose has fairly good rebloom and is quite willing to climb. If you want a headstart in getting it to a large size, I would suggest buying from Antique Rose Emporium in Texas. The shipping can be expensive but that is because they send out such big plants. One of my favorite vendors.

Comtesse O'Gorman is a hybrid perpetual that has a heavy, dreamy scent and climbing tendencies. I don't think it is in commerce at present unfortunately. It is red though, not pink.

Cl. Chrysler Imperial is another worth considering, if you can find it. It's a climbing hybrid tea with powerful fragrance and a liking for heat. Very good rebloom. I had 2 that perished in a wildfire in 2003. I have been unable to find replacements. :(

Another suggestion that you can look up on HelpMeFind is Yolande d'Aragon (I've seen it grow much taller than HMF indicates),

Or check out the damask/portland roses at Greenmantle Nursery and give Marissa a call there. She is very knowledgeable.

Melissa

Here is a link that might be useful: Damask and Portland Roses at Greenmantle

I forgot one, Stanwell Perpetual, I guess because I think of it more of as a spinosissima since it has the tiny leaves typical of the class. Flowers are all damask in looks and perfume to me. HMF says it is only moderately fragrant. Not in my garden. Here, the fragrance is quite strong. I have 2 plants of it. One is over 5 feet tall at around 3 years old and showing signs now of getting much bigger. This one is from Heirloom Roses in Oregon and has been blooming almost continuously since January. Blooms are a pale pink that fades to white in intense sun. I have another one of the same age from Rogue Valley Roses. This one is only maybe one foot tall and rarely blooms. A dud.

Melissa

Here is a link that might be useful: Stanwell Perpetual on HMF