... What’s up here. I mean the magic eye is an AC driven stage, so I wouldn’t think ... polarity would matter. ...
Electrons cannot flow from Plate to Cathode.
Tube (magic-eye or otherwise) passes no current unless electrons "boil out" of a hot cathode and get pulled to a Positive-Plate (with a few non-relevant exceptions).
We cannot reverse current-direction through a vacuum tube, so think of them first as "diodes" with one-way current.
We only get "AC" when our one-way current gets bigger/smaller, and we strip off the large DC part of that changing-current with a blocking capacitor.
So from an initial thought, polarity
always matters.
When I plug the unit into mains, the optical magic eye indicator only responds with the heathkit T3 is antiphase plugged in.
(Dmm in AC, across the T3 and a grounded device, shows mains potential across units)
I wonder if you removed the "death cap" (0.05µF to ground at the PowerSwitch on the schematic).
That cap strips off RF noise riding on the line, and puts X
C = 1 / (2 x π x 0.05µ x 120Hz) = ~26.5kΩ between one side of the line and Chassis.
If you added a 3-prong cord and removed that cap
the "other side" of the line should theoretically be bonded to chassis/Ground at the service panel (is it?

).
I am coming around to the nothing that "Death Cap" is a bogus-warning in the era of plastic caps (of the mid-60s blue "Ajax" caps in Fender or later), but that paper-dielectric caps and a few very early plastic caps cannot be trusted.
Modern Class Y caps may be "metallized paper" but they are rated for safe-failure when connected from Line-to-Ground (they are guaranteed to fail-open, and never fail-short).
So that's one aspect that might be relevant to your observation. The other is:
The unit is a kit build ... 2 prong similar blade power cable.
When I plug the unit into mains, the optical magic eye indicator only responds with the heathkit T3 is antiphase plugged in.
(Dmm in AC, across the T3 and a grounded device, shows mains potential across units)
So with mains polarity aligned, T3 doesn’t illustrate readings optically ...
... I wouldn’t think mains ac polarity would matter.
Should I just swap PT primary for the ... transformer? ...
I hadn't read the manual, but I see there is a "3rd Setting" that disconnects the Audio Input from the 1629, and instead connects one side of a PT winding (whose other-side is connected to
Diodes on the 12C8. At a glance, this appears to be a "diode detector" for
Line Voltage, where the resistors and 0.05µF cap give a slowly-changing DC Voltage that is proportional to the Line Voltage (governed by the Pot; maybe a "Wattage" Pot?). This DC Voltage is applied to the 1629's triode-grid, so once again, polarity matters.
But I see a bunch of primary wires on the power transformer, and they're all "Black"... If you still have the 2-prong cord, you would just flip the cord over to make the magic eye work. If it worked for audio, but not for what I assume is "Watts" then you would swap Black wires on the second-primary to get that feature working.
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When I plug the unit into mains, the optical magic eye indicator only responds with the heathkit T3 is antiphase plugged in.
(Dmm in AC, across the T3 and a grounded device, shows mains potential across units)
Now that I think for a second... vintage test gear never assumed the chassis was "0v relative to earth" and usually did not want the "test gear chassis" to be connected to the "device under test chassis." Inputs were typically floated off the chassis, and yes, there could be significant (even dangerous) voltages between one item connected to Line Voltage and another item connected to Line Voltage. (Reminds me also of the use of "Line Isolation Transformers" that intentionally divorce a tested piece of gear from other line-connected devices, so voltage-differentials like this can be squashed while keeping a 3rd-prong connection to Ground at the service panel).
You can try swapping PT primary-wires, but there's a chance this is "just the way it is" (I haven't really thought enough about it to know whether the chassis voltage-differential would go away).