I must point out ONE case where 'heater' connection matters. Beach radios had battery heated naked filaments. THESE have + and - marks on the filament pin-out. Because the heater is also the cathode, and has 1.5V of DC drop along it, you must connect heater battery correctly to get the designed grid-cathode bias.
There is another larger class of naked filament tubes intended for AC heat. 2A3, 300B. These are typically power tubes with *large* grid swing. Hum is a problem. You center-tap the AC so the average cathode potential is near zero. Also these are usually low-voltage filaments and high signal voltage; like 2.5VAC (1.25V each side) heat and 30V grid drive.
All of these filament tubes fell out of favor with the introduction of indirectly heated heater-cathode tubes. Another name is Uni-Potential Cathode. The filament is bunched-up and slid inside a Nickel cathode sleeve. What happens in the filament
stays inside the cathode, has "no" effect on the operation of the valve. This allowed AC heat of sensitive tubes; also avoids a maze of C- bias batteries (because we can cathode-bias without interacting with heater power).
Hemmingway said a writer needs a good BS detector. Internet writers mostly do not know Hemingway's teachings, and do spew BS. Be careful where you step.
Some sources are low-BS. The now-dead men of RCA and GE and Philips knew their stuff (their jobs depended on it). The tube books do not indicate a 'phase' for the heaters of the tubes we use.
Some sources, listen but verify. Electronics mags from 1930s-1960s published a lot of material, sometimes hum tips. Yes, sometimes the authors were résumé-polishing, and sometimes editors were not practical engineers, so don't trust too deep. And I don't have a bibliography of hum articles. But in my massive reading I do not recall any "heater phase" suggestion (in context of the tubes used in g-amps). Valley Wallman 1948 has nothing on the topic.
An interesting side-note: most 1950s+ 12AX7 use helical heaters, most 6L6 etc use folded heaters. Both are tightly packed into the cathode sleeve. I can't see how either type has a "side", or how to correlate folds against twists to say "this is the same 'phase'".
Another path is to try to logic it out. "Cancellation" implies two equal but opposite signals. It seems unlikely that hum in a low-level stage and hum in a high level stage can cancel without added fuss. Especially since the gain between is usually variable (volume and tone knobs). We do observe hum-null in some amps: there is hum in second stage with Volume full down, as we turn up to maybe "3" the hum reduces (1st stage hum balances 2nd stage hum), then at higher Vol setting the 1st stage hum dominates. Sweet, except this Vol setting is rarely the gain we want to work at. (The opposite case, hum worst at half-way, is electrostatic leakage into the wire from the pot wiper, not heater hum 'phase'.)