It could be the antenna is tuned with the HR2510 in line with specific jumper cables, etc. Certain antennas like the A99 and others are not decoupled very well from the feedline and the feedline and its length can be part of the tuned antenna system. Maybe when the Icom 706 is in line it has a different jumper cable length or something else is different in the coax between the radio and antenna when radios are switched?
I've seen this before when the coax, jumpers, wattmeter radio, power cables and power supply cables can interact a little with the SWR because its all connected to the shield of the coax and hot with RF. If this is the problem I would add a good 1:1 choke balun near the antenna and retune the antenna, then all variations of radios and coax should all read the same SWR.
Another potential problem could be one radio has a problem and its putting out lots of harmonics that are not far down from the main frequency. That would show up as some reflected power because it would be in the 50MHz range hitting an antenna in the 28Mhz range and reflecting back. Something would have to be really screwed up for that much harmonic energy to come out of a radio.