Seem to remember 56K modems won't work back to back, at least not at 56K, as they rely on some trickery with the digital telephone exchanges so you might need to persuade them to connect at a slower speed.
Yep. Pre-56K, your ISP had banks of modems, and when you called, you got a switched circuit (in theory; read on) to the ISP's incoming phone lines, where a free modem would then take the call and connect. It was very much like how a pre-VoIP call center worked, with banks of operators waiting to take your call.
But of course, we haven't actually used
true circuit-switched lines for phones in most places for decades. Instead, at the local exchange (where your phone line physically terminates), an A/D converter would digitize your audio, compress it down to 64 or 56Kbps, and then send it digitally through the phone network, and then at the local exchange at the receiving end, it would be converted back to analog.
Of course, the A/D conversion and compression algorithms were designed for human voice, not for modem noises (or music, which is why hold music sounds so awful). Somebody realized this is silly for a modem; you were taking digital, converting it to analog, converting it to digital, compressing it, decompressing it, converting it back to analog, and then converting that back to digital!
Enter the 56K modem.
In essence, the 56K speeds worked by not actually connecting the client modem through to a modem at the ISP. Instead, the ISP modem is actually at the local phone exchange, and from there it went digitally to the ISP itself. By avoiding the voice-centric ADCs and compression algorithms and instead using the 56Kbps channel bandwidth for the digital data directly, it increased bandwidth and slightly reduced latency. (56K was chosen because not all phone networks supported 64Kbps channels — some used 8Kbps for signaling, leaving 56Kbps payload.)
The other advantage is that only the "last mile" (aka the local loop: the wire between your phone jack and the local exchange) were analog, compared to potentially much, much longer on a true analog circuit-switched line, where it might be many miles of analog signaling between you and the ISP.
As an aside, ISDN essentially connected you directly to the 64Kbps digital channel. The total absence of A/D conversion is why ISDN was a lot faster for Internet use, despite having similar bandwidth: far less latency.
ADSL (ADSL, VDSL, etc) works the same way as 56K modems, in that the ISP equipment (the DSLAM) is actually in the phone exchange. The only-the-last-mile-is-analog advantage applies here, too. In fact, more so, since ADSL depends on using
only frequencies beyond what POTS lines were designed for (since ADSL leaves the voice frequencies free so that it can piggyback on an active voice line). POTS lines' frequency bandwidth plummets with distance, taking data rate "bandwidth" with it. Because of this design, the uplink channels are in one block of frequencies, and the downlink channels are in a different, much larger block of frequencies, so the modem can only send one block of frequencies and receive the other, and the opposite for the DSLAM.
In contrast, the much rarer SDSL (SDSL, SHDSL, etc) (which doesn't piggyback on an active voice line, and uses voice frequencies) can theoretically have the endpoint in the local exchange, but was more commonly used with a leased pair (a true pair of wires permanently wired through from you to the other end, onto which you can place whatever signal you want, within reason). As such, SDSL modems must be able to both send and receive all the same frequencies. Consequently, two ADSL modems are categorically incapable of directly communicating with each other, while two SDSL modems will happily connect over a copper pair.
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