In reply to Lankyman:
That's a shame. I was there in 2007 I think, so suspect things may have changed. Found this online (tripadvisor, of all places - which makes sense when you read the last sentence):
Superb walk, certainly one of the best in Majorca. We climbed up from one if the car parks off the main road going through Valldemossa. There is also parking on some of the residential roads near the sportsground, closer to the start of the path.
After travelling along the path for about 20 minutes, you arrive at a junction with the main route seemingly going around to the right. The path on the left goes past a wooden hut; when we visited it was manned and walkers were not allowed to enter without a permit (apparently only a few are allowed to go through each day, apply for a permit in advance). This left hand path leads to the Mirador de Can Costa where the Archduke had a viewpoint built, protected by a stone wall. The right hand path winds its way up to the Archduke's path at the top of the ridge coming out near Caragoli, one of the peaks. Access is free, no permit required. The easy to follow path climbs to an airy viewpoint, below the main ridge/plateau. The path (now less distinct) then climbs through the trees and up onto the plateau itself - several routes appear to cross this relatively flat area - keep heading forward until you reach the Archduke's path which runs along the northern edge of the plateau.
I would recommend that you turn left at the Archduke's path - this is the most 'interesting' and attractive part of the walk. Caragoli, the main peak on this section is a short distance away, not long after on the right is a path that drops down to Deia. Continuing along, the archdukes path becomes a ridge walk with a precipice on the right and the land dropping away (less steeply) on the left. Note the path itself is broad and well maintained. A final summit is reached where the path climbs, somewhat reminiscent of Jacobs Ladder, into the heavens. Looking back, it is possible to see the path winding its way back towards Caragoli. Ahead is Talaua Vella with a mountain hut built at the time of the archduke just below. Continuing, the path drops down to a hermit's cave, which is worth a look, if unoccupied.
The Archduke's path then goes across a wooden style above a stone wall, however there are warning signs that the land is private with restricted access. Personally, I would have liked to continue across the style and onto Talaia Vella and the later viewpoint of Mirador de Can Costa. However, I instead followed others in front down a path which was still under construction. The path was less distinct and after about 10 - 15 minutes, goes down a rather difficult steep rocky section. The route then winds around and comes out on the path climbed earlier in the day, above the wooden hut. I found this last section disappointing, the authorities need to ensure that the full length of the path is available to all to enjoy.