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):
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):
- 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
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.