Is there any "hack" to determine the exact itinerary of an Orbitz Bargain Fare flight?

We might assume that Orbitz has some contract terms with some airlines to offer extra discounts in some cases, and doesn't want to let people know the "secret" booking in advance of purchase, to force people to purchase from Orbitz, and prevent people who care about the airline and routing from taking advantage of the fare. So it might not be possible to determine the exact itinerary they are offering, but it might be possible to make educated guesses.

You know some of the parameters, including dates, origin, destination, time range, and range of stops. You can use what you do know to narrow down the possibilities, and then check flight loads and published fares to try and guess the likely routing.

I'd suggest checking ITA Software's Matrix search as a starting point, since it's free and does a pretty good job, especially when there aren't an enormous number of possibilities. The main downside of Matrix is that it has an internal timeout limit, so if the search has too many possible flights, some of them won't be searched. In this case, since you know the exact dates and the exact departure and arrival airports, and departure time ranges, and a limit for the number of stops, you can restrict the number of possibilities that Matrix needs to search.

Here's a screen shot of the Matrix round-trip search form:

Fill in the exact airport codes for departure and arrival, the exact flight dates, set the "Stops" to the maximum that you know (in your case, "up to 1 stop"), and uncheck "Allow airport changes."

Matrix will show you the lowest cost options it can find within the maximum time it internally allows.

The lowest cost option that Matrix finds is a candidate for what Orbitz has. I would also use ExpertFlyer, although this is a subscription service (with a free trial period). I'd first do a Fare Information search in ExpertFlyer to see the lowest published fares for the dates. It returns base fares without taxes and fees, so they won't be directly comparable, but you can compare the lowest published fare with what you found in Matrix and Orbitz.

Here's a screen shot from ExpertFlyer's home page showing the search options. "Fare Information" is the next-to-last one:

Here's a screen shot of the Fare Information search form. Be sure the "Airline(s)" field is empty and "Validate Fares" is checked.

You'll see a list of fares, with the cheapest at the top. The Airline column shows the airline. The Effective Date, Expiration Date, Min/Max Stay, and Advance Purchase Requirements column show important information about restrictions that the fare has. You can click the third from the last column at the end to see the fare rules, or the second from the last column at the end to see the routing restrictions for the fare. The Fare column shows the base fare (not including taxes or fees). In this search, Wow Air has the cheapest fare, an Instant Purchase online-only fare that is non-refundable:

You can click the "View Flight Availability for ..." link at the top of the Fare Information results to search for flight availability. In the Flight Availability search box, you can fill in the airline code from the Fare Information results to restrict the search to that airline.

Unfortunately, you can't restrict the maximum number of stops in this search, but ExpertFlyer sorts by the number of stops:

In the list of flights, you can see the number of connections, the operating airline, and the available inventory. Flights that have the largest number in the lowest inventory (the last inventory code shown) are the best candidates for the Orbitz fare, since these have the most seats for sale in the cheapest inventory.

However, ExpertFlyer can only show what is published in the Global Distribution Systems, which almost all airlines use. There are some airlines that don't participate (for example, Southwest), so those airlines' fares and flights won't be shown.

In this example, the cheapest fares were on Wow Air, but the only available flights show as two connections, whereas your Orbitz results showed zero or one connection. When this happens you can redo the Flight Availability results, either removing the Airline(s) to see flights for all airlines, or filling in the next-cheapest airline you found in the Fare Information results:

Leaving the Airline(s) field blank shows more possibilities:

As before, the flights with the most seats for sale in the cheapest inventory are likely candidates.

My approach to this is based on the taxes and fees (YQ). Government taxes are refundable so Orbitz shouldn't be able to fake those. However, they seem to be doing something very dodgy. It would help a lot if someone had an actual example of a fare that was bought and showed what they flew and what the fare and fee breakdown was.

Here's an example: SFO-JFK May 28 (2 days time), one-way.

Here we see a fare of $224.46 + $26.20 fees

US domestic fees can be seen on Delta's site for example Tax is 7.5%, Segment fee (ZP) is $4.10, Security fee (AY) is $5.60 and Passenger Facility Charge (XF) is up to $4.50, however, it's $4.50 everywhere as fare as I've observed.

If you look at the cheapest direct fare on Orbitz for that itinerary, you'll see:

$43.64 is 7.5% of $392.56 + $5.60 + $4.50 + $4.10.

If you look at the multi stop itineraries you'll see, for example, a one stop flight with 2*ZP+2*AY accounted for in taxes and fees.

Here's the problem. The example Orbitz bargain fare above must include at least 7.5% tax, + $5.60 + $4.40 + $4.10, but it doesn't(!), nor do any other of the three examples I found.

The $224.46 fare should have $16.83 tax + $5.60+$4.50+$4.10 which comes to $31.03 in total taxes which means $4.83 has disappeared. The passenger service charge is not fixed, so the only conceivable legal possibility (to my non legal mind), is that Orbitz has negotiated a Passenger Facility Charge lower than any other airline, even Spirit.

The same sort of analysis can be performed on international flights, where the majority of the taxes/fees is made up of YQ, which used to be called fuel surcharge when it suited the airlines, and is now typically called just a surcharge or a carrier surcharge or some arbitrary nonsense to allow the airline to charge fees on redemptions.

I leave that part as an exercise to the reader :)

My other observation is that there is no other observable fare with the same cost, meaning that Orbitz has negotiated consolidator fares with the airline that will not be found on a GDS.