Electrical – Re-use circuit breaker handle tie

It sounds like you may be mistaken as to how this is wired, or that perhaps I'm just not understanding your explanation. As others have mentioned, it's not possible to get 240 volts from a single pole in a 120/240V split phase system. Each tandem breaker provides 2 120 V circuits, this is true. However, if you measure between the terminals on a single tandem breaker, you'll get 0 volts. This is because the terminals are both powered from the same leg, and so are at the same voltage potential. If you measure from a terminal on the top tandem breaker to a terminal on the bottom one, then you'll measure 240 volts. This is because each breaker is connected to a different leg, which are each one half of a 240 volt circuit.

With all that said. For this setup to work, one appliance would have to be connected to both breaker. Something like this...

Notice that each appliance circuit has one wire connected to each of the tandem breakers. In this situation, you'd need a device like Speedy Petey shows.

Which ties the breaker handles together, to provide common trip characteristics.

Notice how the inner handles are tied together, and that the outer handles are also tied to each other. This way if either trip (or are turned off by the user), the entire circuit is shut off.

If this is wired the way you've explained, where the dryer is connected to the top tandem and the heater is connected to the bottom. Then there's some magic going on in those breakers.

The Eaton type CH is typical among modern breakers in that multipole common tripping is performed by an internal mechanism, not by the handle tie. This is noted in the Eaton catalog in the spec description of the type CH breaker (page V1-T1-31 of the Eaton catalog, emphasis mine):

Product Description

Quick-make, quick-break switch mechanism combined with inverse time element tripping operation and trip-free handle design. Type CH circuit breakers trip to the OFF position, eliminating nuisance callbacks. The CHF family also includes a trip flag to differentiate between a trip and the breaker being turned off. The thermal-magnetic trip curve avoids nuisance tripping on mild overloads while reacting almost instantaneously to severe short-circuit conditions. Multipole breakers have internal common trip connection to operate all poles simultaneously. Handles are marked with ON-OFF indication and ampere rating of the breaker.

and in footnote 2 on the accessories table (page V1-T1-38):

2. Handle ties: typically used to join two similar independent single-pole breakers to form a two-pole noncommon trip breaker.

This means that the combination of 2 single pole type CH breakers and a handle tie is only usable for 240V only or 120V only circuits as per NEC 240.15(B)(1) and 240.15(B)(2):

(B) Circuit Breaker as Overcurrent Device. Circuit breakers shall open all ungrounded conductors of the circuit both manually and automatically unless otherwise permitted in 240.15(B)(1), (B)(2), (B)(3), and (B)(4).

(1) Multiwire Branch Circuits. Individual single-pole circuit breakers, with identified handle ties, shall be permitted as the protection for each ungrounded conductor of multi-wire branch circuits that serve only single-phase line-to-neutral loads.

(2) Grounded Single-Phase Alternating-Current Circuits. In grounded systems, individual single-pole circuit breakers rated 120/240 volts ac, with identified handle ties, shall be permitted as the protection for each ungrounded conductor for line-to-line connected loads for single-phase circuits.

As to breaker ratings, the CH line has you covered as well. All standard (i.e. no GF or AF functions) single pole CH, CHF, CHT, CH-HID, CHP, CH-M50, CH-HM and CHP-HM breakers are rated 120/240VAC, so they are listed for handle tied service on 240VAC only split phase circuits.

Furthermore, the UL Circuit Breaker Marking Guide states in paragraph 40 that

An external handle tie alone does not qualify as a common trip mechanism — a breaker of this type is marked to indicate it is an independent trip breaker.