Yup, thats pretty much it - most common slip is your feet sliding out from under you so if your axe is on uphill side you are in the right position to put your weight over it and self-arrest, or often stop yourself before you even start sliding. On the downhill side you have to roll over/spin around first (use the axe as a pivot point to spin) which takes time and lets you gain speed.
I recommend having a practice session on a slope with a safe run-out before hitting steep terrain - practice falling and self-arresting from every angle you can think of (feet first, head first, upside down, sideways...) so you know what to do if it happens accidentally later. Sometimes in soft snow you can get better braking force from the adze rather than the pick, takes confidence to pull it out and turn it round while you're sliding down the hill though...
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