After several years of uncertainty, there is now an established trailhead at the Top of the World Drive, which is off of Wasatch Boulevard and not far south of the mouth of Big Cottonwood Canyon. You can park on the street, and then head up the dirt road, which eventually turns into a trail before heading up Ferguson Canyon.
There is one reason I can think of to climb this route. This is not an easy route, and it is very long, but this is the safest (but not the easiest!) route I know of for winter ascents (more recently, Mark Thomas reports that the North Ridge of BF Twins may be an equally safe ascent). The words "safer" winter route is used in a realitive term.
The Broads Fork, Stairs Gulch, and Tanners Gulch routes are extremely dangerous in winter with many avalanches.
July 1st, 2005 - The Broads Fork Twin Peaks seen from the valley. The Hounds Tooth Ridge ascends the skyline left of the summit.
First follow the very steep game and hikers paths up to Hounds Tooth Rock. Pass the rock to the right, and scramble along the ridge all the way to the summit of Twins. This is a strenuous route, but it is probably the safest route to Twins in winter. No detailed route description is needed, just takle the ridge all the way to the summit. Above the timberline, the ridge is very rugged and it curves south to the summit. Most of the ridge is class 2+ to 3, but with a few class 4 sections. The very last section of the ridge is the same route as the Robinson Variation
from Broads Fork. I would suggest two days for a winter ascent with a camp somewhere above Hounds Tooth. The route can be done is summer too, but it is more difficult and strenuous than the Broads Fork route, and would be very hot down low.
This is along the final section of the route.
In winter bring an ice axe, crampons, and winter climbing and camping gear. Avalanche beacons and shovels are recommended as well.
WARNINGThis is not an easy route. The words "safer" winter route is used in a realitive term. If corniced, this could be a very dangerous climb. Stay off the ridge if cornices are present.
The Hounds Tooth Ridge Route climbs the ridge that divides the shadow and light and passes over the rugged peaks visible above. This is probably the "safest" route of Twin Peaks in the winter, but it is not easy. It still could be dangerous as well, so use extreme caution.