I never get my HR Zones to line up with my Power/Pace zones. I use power exclusively on the bike and watch my HR to evaluate fitness or how I’m feeling for the day. I use Pace for key run sessions. On easy days I’ll keep my HR within the desired zone and watch Pace to adjust. Swimming I don’t bother, just use the clock. There’s just to many variables that influence HR. Your Threshold HR should be different for the Run and Bike.

Don't really know what my swimming HR is as I don't have the tri strap so don't get any kind of reliable HR reading until I'm getting out of the water (at which point it's pretty high but that could just be the adrenaline rush of hitting T1). HR is different for running and cycling. E.g. In running I typically average mid to high 160s for long race efforts including marathons and 70.3 runs (IM run is a little lower). In cycling I only see that HR at FTP efforts or higher and it's not sustainable beyond an hour. Highest HR I've ever averaged in a tri bike leg was mid 150s, and even then my run suffers. At 85% FTP my HR is typically somewhere in the mid to high 140s.

I always assumed it was because running used more muscles than cycling so sucked up more oxygen at any given effort. My easy pace HR is also ~10bpm higher in running than cycling.

Interesting question

From my experience and observations and readings, HR zones for each sport can differ, based on your experience/strengh on each sport.

For me, coming from cycling :

Bike : my HR zones are "classical", with FTP HR at 158 BPM, max HR 182 BPM

Run : also using the legs, my HR zones are similar, but I can't handle as much duration and intensity as my running is more recent. If I push as much as cycling, I will develop injuries.... so I limit intensity and durations, according to BarryP principles.
So, same zones, but I'm "under-using" the engine to save muscles and tendons

Swim : recent to swimming, my arms muscles are small and weak. So my HR zones associated to "arm muscle effort" (or Lactate level) are lower.
My FTP HR is probably around 130 (instead of 158). More or less is overusing or less using legs.
It moves up slowly with my training.

Swimming HR should be less at relative intensities. The reasons: swimming is horizontal, meaning the heart has to work less to pump blood to the brain and up from the legs; hydrostatic pressure in the water squeezes blood toward the core (another reason why you have to pee more when swimming); swimming uses smaller muscles than running or cycling.

I knew my HR zone should be less for swimming. I thought it was because you are horizontal so your HR works less in that position.

I remember seeing Sanders post that bike HR was ~120 and run HR was ~150 in a race. I was surprised to see that much of a jump on the run as I typically try and keep a low z2 on the bike and high z2 on the run (with an overall variation of 10-15bpm).

Are you trying to prescribe work rates based upon percent HR? My approach would be to measure your maximal heart rate during each of the disciplines....assuming that you're healthy enough to undergo maximal exercise tests. Without each discipline's known max HRs (and aside from perceived exertion), an absolute HR value is entirely arbitrary as it relates to relative effort.
Also, you should be using Karvonen's method (heart rate reserve) if you're interested in setting up your HR zones; not percent of max HR.

Edit: also, there are several really interesting physiological mechanisms (both central and peripheral) that are differential influenced during swimming vs. running. You should read about them if you're interested.

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