All Yamaha DX9 patches are parameter compatible with the DX7. That is, you can issue a MIDI patch dump from a DX9 and it will quite happily load into a DX7. However the DX9 has 4 operators while the DX7 has 6 and the DX9 has just 8 algorithms compared to the DX7's 32. So, how can all of those parameters map automatically?
Here's what's really going on.
The DX9 is essentially a DX7 with Operators 1 and 2 turned off in firmware. With those two missing operators in mind, we can look at the DX7's 32 algorithms and see which of them still "make sense". That is, which of the 32 algorithms have operators 1&2 arranged so that they are either 2 independent carriers or a single modulator-carrier pair that can be removed without affecting any other operators in the algorithm?
It is these DX7 algorithms that are implemented by the DX9 and the internal algorithm number used is the same as that of the DX7. This is why DX9 patches can be loaded into a DX7 with no alteration.
(click on an image below to step through the DX7 to DX9 algorithms)
So, there are anomalies, the first of which being Algorithm 3, the only algorithm that has feedback on Operator 2 rather than Operator 4.
On the DX7, feedback could vary across multiple operators, however within the DX9 algorithms; feedback is confined to either Operator 4 or 2.
The next generation of 4 Op synths removed this 'anomaly' and feedback only ever appeared on the last operator (4). As a result, although it is functionally identical, the operators in Algorithm 3 are configured differently on the DX9 compared to the later 4OP synths. (which is something to be aware of when converting patches).
The second anomaly is the missing algorithms, There are 3 algorithms (13, 25 and 28 on the DX7) that could be implemented by any 4OP FM Synth and It is not clear why they are not included in the 'standard' 4OP algorithms, feedback is on a different operator for algorithm 28, however there doesn't really seem to be any real reason to exclude them.
It could just have been a decision to further distance the DX9 from the DX7 by restricting the algorithms to just 8, or a case of picking the closest "power of 2' number to the available algorithms,.
Interestingly enough, the recent Yamaha DX Reface has 12 rather than 8 algorithms and those same 'missing algorithms' 13, 25 and 28 appear on the Reface as Algorithms 2, 10, and 6 respectively.