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Does the winding direction/handedness in each side of the T507 binocular core transformer have to be the same.? Hans stated in his 2023 Evolution in Radio Design paper (pg 21) that:

"... Here we use a more compact method, a
binocular core. The two transformers are each wound in the two
separate holes. As far as the binocular core is concerned, what happens
on one side is electrically invisible from what happens on the other. So
the use of a single binocular core instead of two toroids is a parts count
reduction benefit and also in our case, crucially, helps reduce the space
required, which in QMX Is at a premium."

?

I've wound one side (10 turns) and started the second but I think I've reversed the direction and want to know if I have to start it over.

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BTW if there is a trick to getting these windings tight and neat I haven't figured it out.? My other torroids turned out OK looking but this one is not behaving.? Probably does not matter but I would prefer it look pretty (pilot's axiom - "If it looks right it will fly right").

a) no the direction doesn't matter, but I always do mine symmetrically

b) I use my finger to press the winding around the ends of the core, bending it so the wire sits right up against the ferrite

I redid my windings as you suggested and they came out better.? Not perfect but better.

After soldering in place, when I checked the connections with a DC meter on ohms everything seemed to be very low resistance and very close to the same readings.? I've looked at the circuit diagram and can see that it is basically a few short lengths of wire more or less connected to one another and ground so I want to convince myself everything is OK.? But ....

Is this normal?

Yes, all terminals of T507 are at DC ground.? The inductors allow the RF signal to pass, and generate the FWD and RVS signal detection for power and SWR calculations.