The Optum RX mail-order pharmacy has caps where both sides can be the "bottom" and screw onto the bottle, but one side is childproof and the other is just a normal screw attachment. Since they come with the childproof side of the cap engaged, this does present the bootstrapping problem of getting the bottle open to begin with, but thereafter you can just use the other side of the cap.
They're pretty common. I had a sinus infection earlier this year that wouldn't go away and had to get some antibiotics. I picked up the RX from a local Walmart pharmacy (closer to where I live) and they use the same double-sided caps.
But again, it's like the other OP said: The childproof side is the one that they put on first which could present an issue for older people. I'd imagine you could ask them not to do that, though. I never really thought about it.
The Optum RX mail-order pharmacy has caps where both sides can be the "bottom" and screw onto the bottle, but one side is childproof and the other is just a normal screw attachment. Since they come with the childproof side of the cap engaged, this does present the bootstrapping problem of getting the bottle open to begin with, but thereafter you can just use the other side of the cap.