Dam breed A, sire breed B doesn't equal dam breed B, sire breed A?

I got confused yesterday, reading about quarter horses. It says quarters are
Descendants of arabian, iberian, etc. stallions and thouroughbred mares.

Why is it different if dam is tb, sire an arabian and if dam is an arabian and sire a tb?

What exactly do you mean?

I'm asking if you get a different horse when you breed tb mare and arabian stallion than if you breed arabian mare and tb stallion?
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No. Any first generation foal from a TBxArabian cross would be an Anglo-Arab.

They would both be classed as anglo arab but sometimes having a specific breed as the sire at the top over the dam can make a difference to what the characteristics that result.