Wiring a Split Outlet with Two Neutrals
That's not a weird neutral.  That's a switch loop.  

They are correctly using the white wire for always-hot, precisely to invoke that "WTH" response.  By using white as always-hot, it assures the white wire will always light up a voltage tester.    If white was a switched-hot, it might not always light up, and could be mistaken for neutral.   

What they failed to do, however, is mark the white wire with black or colored tape, as is required today (also) to designate it as a hot.  

You are welcome to retrofit grounds to help protect this circuit.  However, better protection would be achieved with GFCI.  You can't fit a GFCI+receptacle combo device here because they don't allow splitting.   However you can fit one at a receptacle upstream if there is any, or a GFCI+circuit breaker combo device if there is not.