Our vestibular organs detect acceleration; they’re biological accelerometers.

Hence my singling out the inner ear, which contains the vestibular system.

“Uneven acceleration” would be what, nonzero jerk or jolt or one of those derivatives, and it would be detected as a change in acceleration.

By "uneven acceleration," I'm referring to acceleration that varies with respect to position within your body at any one moment in time, not jerk or any other position-time derivative. The only reason your vestibular organs detect acceleration at all is because the fluid in your inner ear does not accelerate in line with the semicircular canals that contain the fluid. The canals, attached to the rest of your body, move first, and the fluid, which flows freely, tries to stay where it is as the canal moves around it. Since the canal is a closed system, the fluid must eventually yield and reaches an equilibrium where it pools up at an angle that directly transmits the accelerating force from the canal to the fluid. Change in acceleration, or jerk, is perceived when the fluid changes position. Constant acceleration is perceived when the fluid is in an equilibrium position that isn't the same as the position it's in when you are stationary.

In both cases, my main point was that it is the capacity of various parts of your body to not accelerate rigidly with each other that cause the perception of velocity, acceleration, and jerk. For all practical purposes, it is enough to say that our body detects acceleration. More strictly, however, we have to understand that it's not the acceleration itself we are detecting. It is the discrepancies between the acceleration of various parts of your body and the different equilibrium points that allows that detection.

Acceleration due to gravity is no different; if you’re upside-down, you know it because gravity is accelerating you.

Due to the nature of relativity, there is a big issue with semantics here. If you are stationary (or at a constant velocity) with respect to a gravitating body (e.g. Earth), you are accelerating with respect to space-time. If you are stationary (or at a constant velocity) with respect to space-time, you are accelerating with respect to the gravitating body. I suppose you could label both as gravitational acceleration, but for consistency's sake, we almost always use the Earth (or whatever the local massive body is) as the reference frame, and only say you are accelerating due to gravity when you are falling, in orbit, accelerating down a slope, etc.

If you are upside-down and supported by something rigidly attached to the Earth, it is the Earth and that structure that provide the forces necessary to keep you accelerating with respect to space-time. Again, it is the uneven propagation of the forces from the contact points through the rest of your body that allows you sense the pull. Remove these forces and there'd be no sensation of acceleration.

At the orbit of the International Space Station, the gravitational acceleration is still quite high at 0.88 g. The astronauts on-board, while I'm sure they know they are undergoing acceleration, feel none of it.