I believe it's a combination of both. Inhaling the fumes would cause the production of sulphuric acid in your sinuses, causing eye and nose leakage. However, the amount of water normally present in the sinuses is greater that that on the surface of the eye, so any acid produced would be more dilute, and take less flushing out to return the pH level to normal* . The acid produced in your eyes is obviously not concentrated H2SO4 (i.e. it doesn't transform your eye sockets into steaming holes of raw flesh),so that which is produced from inhalation is likely to cause an even weaker reaction. Hence, achan's goggle idea, while it may not completely stop eyewatering, would at least cause a decrease in the severity of the response.

Actually, I noticed several years ago that wearing contact lenses massively decreased my onion-tearyness, and made the rest up from there. Sounds logical though.

* I have no clue as to what normal nose pH is, and very little inclination to shove test paper or a pH electrode up my nose. Anyone wishing to take part in a worldwide survey of nasal pH should feel free to /msg me with their results.



ToasterLeavings will be attempting a nasal pH reading using a fish tank test kit, just as soon as 15mL of nasal fluid become available. More information will be provided as it comes to hand.
If I get a few more volunteers, I'll send out pH strips and node the results.