Why are intermediate cardinal directions read differently in Chinese?
In Chinese,

East/West comes before North/South: e.g. the phrase 东西南北
East comes before West
South comes before North: e.g. 南北朝, 南拳北腿

In asking why these particular orders, we're begging the question of why the Western cardinal directions are in their order? That is:

North/South comes before East/West
East comes before West
North comes before South

After examining reasons for both, hopefully it will become clear why the orders are the way they are.
There are two differences to consider:

whether the East/West axis comes before the North/South, and
whether North comes before South

Warning: there's a lot of guesswork here; if anyone has better reasons or evidence please contribute, which is why I'm making this a CW answer
Why is East/West more important than North/South?
If you were a prehistoric human and observed your world, the sun would be one of the most notable features of your world. It would be easy to notice that every day, it moves from one side of the sky to the other: East to West.
So it should be easy to see why East/West is so important, and why East comes before West. There is evidence that in Western cultures, East once had primacy too, especially in the Medieval period, where most maps were East-oriented. It was also related to the phrase &quot;to orient&quot;, where orient also means East.

Why did North/South become more important than East/West?
This leads to the question of why did North/South become more important. The answer is simple:

The compass's influence cannot be understated - it isn't one of the four great inventions of ancient China for no reason - as it was immensely useful for navigation and seafaring. North/South became more important in China too, at least for navigation and cartography, as evidenced in this map dating from the Ming dynasty:

Why is North more important than South, and vice versa?
Which leads to the final question, why North is before South for the West, and South before North at least in the Chinese language. This may be the hardest question to answer, so let's go back to first principles, look at the reasons why North became important in the West, and see if those same reasons or different ones are at play for China:

Compasses point North (but they also point South!)
The North Star, Polaris, is visible in the Northern Hemisphere; no stars pointing South would be visible
Egocentricity: people associate up with good, so naturally they'd orient themselves at the top of maps relative to their neighbours. In Medieval times, the greatest enemy to Europeans were various Islamic sultanates, generally located to their South.
Ptolemy oriented his maps North; in his time the most interesting places were in the North which may have influenced his decision.

What about China then?

For most of China's history, their greatest external threats were various nomadic cultures to their North. Look no further than the walls they built for evidence of this.

These reasons are not that convincing, but hopefully they are a start. It's interesting to see that Chinese maps were once South oriented, but eventually they became North oriented too, leaving the word ordering as a legacy of ancient times.