Interesting idea.
I think the potential downside is that your emergency fund will just shrink and shrink over time, to the point where even during a downturn you'd barely be back at your cost basis.
In general I think hedging can be kind of pointless (why wouldn't you just buy less of stock A rather than buying some of its inverse as insurance?) and my guess is that logic holds up here as well. I think you'd probably be better off holding onto some cash.
However, one thing that has me scratching my head is the idea of investing in a dividend play like HYG while holding its inverse, SJB.
Given a dividend of ~5% from HYG, wouldn't you effectively have a guaranteed ~1.55% overall return by holding equal amounts? Or is there something I don't know about the nature of inverse ETFs?
Edited to account for .95% fee of SJB.