The second harmonic would have to be really bad for this to happen. If the second harmonic were only 10dB down from the primary frequency and you had 100% reflection back from the antenna at the second harmonic, and you have very low loss coax you would read about a 1.9:1 SWR with an otherwise perfect match at the primary frequency. A higher SWR reading from a second harmonic would mean the radio is putting out a lot of power out of band.
PRCGUY is probably on it with regards to harmonics.
It could be a 2ND harmonic issue with the ic706 when used on cb which is out of its desgned band giving a improper VSWR indication or it could also be a spurious non harmonic issue with IC706 WHEN used in the CB band, what does the 706's spectrum look like at 27 MHZ as compared to 28 MHZ?
When the ic706 is modified for cb use, even at low power output, what does the 2nd harmonic and other non harmonic spurious look like compared to the HR2510?
2Fo should be around -50dBc, when you operate a hacked ham transceiver out of its designed band and you really should look at the harmonics and spurious emissions out of the transmitter with a spectrum analyzer.