Why A linear transponder doesn't demodulate anything; it takes an entire slice of uplink spectrum, mixes it with a local oscillator, and retransmits the resulting difference product on the downlink band. Because the downlink is LO minus uplink, the passband comes out reversed: a signal near the low end of the uplink window appears near the high end of the downlink window, and upper sideband becomes lower sideband. That spectrum reversal is exactly what the word 'inverting' describes, and it also makes uplink and downlink Doppler shifts partly cancel.
Watch out Detecting and remodulating on the reverse sideband describes a regenerative or demodulating repeater, not a linear transponder, which must stay linear so it can relay many SSB and CW signals at once without touching their modulation.
Linear transponder = mix, don't demodulate. Difference product means the band flips: transmit USB up, listen LSB down.