Он нам показал свою машину.
Is that a natural way of saying it or would one only do so with a certain purpose which would be figured by the listener, because it does sound strange compared to
Он показал нам свою..?
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It is OK and only slightly coloured. The most neutral way is indeed "Он показал нам свою машину" but the variant with "нам показал" is your next best choice.
In Russian, while the sentence roughly follows the subject-verb-object word order, it is typical to gather old information closer to the beginning and your statement closer to the end of the sentence. Not a "rule", just a generic explanation of some phrasings that look strange to a native speaker of a language with a more fixed word order (i.e. the order remains the same even if the message is significantly different).
You see, in most cases you sentence has the things you are going to talk about and then what you say about these things. Now, the pronouns ("нам", "тебе", "тут") most likely represent the old information, so as a rule they tend to be closer to the beginning of a sentence than the position a normal noun would feel comfortable in. A second peculiarity is that Russian does not heap up all possible adverbs by the end of the sentence — unlike English, which tends to do exactly that ("He worked very well indeed at the bar yesterday").
In your case I would use the following, in the order of decreasing neutrality:
"Он нам показал свою машину" sounds a little bit emphatic because the pronoun normally goes after the verb. So, this example of yours would be translated as "He showed his car to us (only to us)". The sentence "Он показал свою машину нам" sounds more emphatic.
But sometimes when we speak we don't pay attention to the word order, so in this case when we say "Он нам показал свою машину", it also sounds natural. You have to pay attention to the context to see if it is emphatic of not.
"Он показал нам свою машину" sounds neutral. it is more natural and correct way to say it.