TL;DR hyperventilating for 60 minutes for psychedelic/meditative/therapeutic effect: is this dangerous?
Hyperventilation can lead to cerebral hypoxia, as hypocapnia (reduced CO2 in blood) causes vasoconstriction in the brain along with alkalosis which impairs the ability of the blood to deliver oxygen (Bohr effect). This is all pretty understandable and there's an extremely detailed breakdown of the mechanics and effects of hyperventilation in this excellent article by an anesthesiologist.
As a result of the hypoxia and alkalosis, hyperventilation can have some pretty psychedelic effects (tingling, numbness, spasms, confusion, disassociation, hallucinations, generally altered state). Some New Age practices take advantage of this (under the term "breathwork" ), with a whole layer of pseudoscience and/or mysticism on top, but I've tried it once and the effect is real: hyperventilating for that long produces some crazy results!
Hyperventilation-induced hypocapnia is "well tolerated" but the only concrete info I can find about cerebral hypoxia is that you lose consciousness about 15 seconds after oxygen supply to brain is interrupted, and brain damage can start 4-6 minutes after that. On the one hand, hyperventilation won't even come close to cutting off oxygen supply. On the other hand, these practices can keep this up for an hour. The widely reported "choking/fainting game" has caused injuries and fatalities, but has components that make it more dangerous - strangulation, pseudo-Valsalva maneuvers, and holding your breath.
How dangerous is this? From what I can tell, mild hyperventilation can cause noticeable mental effects without being dangerous (common in panic attacks for instance). But what if you take it really far? What are some sane limits on duration, and how could you identify that you're getting into dangerous territory?
 Via citation on Wikipedia (sorry): Laffey JG, Kavanagh BP (2002). "Hypocapnia". N. Engl. J. Med. 347 (1): 43–53. doi:10.1056/NEJMra012457. PMID 12097540.
Note: This question Is a hyperventilation+hypoventilation breathing exercise (Wim Hof Method) more likely to prevent or promote cancer? has a lot of great info about the effect of hyperventilation (and hypoventilation) on the body, but it's in the context of regularly performing the exercise long term, and the breathing pattern is different. My question is about the possibility of acute damage from pure hyperventilation in one go.