I have seen the phenomenon you mention quite often. It comes from the asymmetrical magnetic system many speakers still have, and it is easily correctable-on the manufacturing line, not by the user-either one of two ways.
D.B. Keele calls the phenomenon "suck-in" or "suck-out" and says that the displacement can be either forward or back, in a vented or closed box.
I have only noticed inward displacement, and only in a reflex box. I'm not saying it does not occur in the other instances, only that this is how I have observed it.
The way the manufacturer can correct this is to:
A) Make the pole piece of such size so that there is just as much magnetic flux in front of the voice coil as behind it;
B) Put in a Faraday ring-otherwise called a "shorting ring".
Both methods are effective. The Faraday ring has the added effect of lowering the speaker's inductance.
In the "suck-in" I mentioned, as the frequency approaches Fb, the speaker quite visibly goes to the end of it's travel, and only moves from it's back end forward and back. Half the wave form becomes clipped off!
Incredibly, it doesn't actually sound that bad. When you clip off half a pure sine wave, you get high second-order harmonics, which are called "overtones". It actually comes out sounding pleasant. But needless to say, pleasant distortion or not what we are going for when we build an enclosure.
The fact that the result, while actually a rather drastic phenomenon, really doesn't sound that bad might well be the reason all but a few manufacturers kind of ignored the phenomenon until recently.
European makes dealt with the issue about 12 years or so, usually the more expensive makes. Peerless, with it's 260 model, was the first popular priced woofer that had a Faraday ring available in a speaker that was well under $100, to my knowledge. This had a lot to do with my becoming a Peerless fan.
Now, a great many woofer manufacturers are paying attention. When you see "Faraday ring", "shorting ring", or "symmetric magnetic drive" and talk of reducing second order harmonics, you know the manufacturer is taking care of this.