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Disclaimer: Swedish native here, but no grammar expert.

Without the preposition, then in most contexts the verb _blanda_ means _mix_. The verb _röra_ can mean either _move_, _stir_, _touch_ (physically or emotionally just as in English) or _affect_. You can more specifically use _beröra_ and then it can only mean _touch_ or _affect_.

It does depend on how you use them as well as the context. The prepositions do change the meaning of the verb. 

- _Blanda om_ = _remix_/_reshuffle_. _Blanda kortleken_ = _shuffle the deck_. _Blanda om kortleken_ = _reshuffle the deck_. No, you can't use _röra_ here.  _Saka_ means discard/sacrifice but only in terms of card games. _Jag tvingas att saka ässet = I am forced to discard the ace._

  _Röra om_ = _stir_. Pretty much a direct translation. Often stirring something physically but like _stir_ it can also be used just as stirring things up metaphorically.

- _Blanda ihop_ = usually means _mix up (terms)_/_confuse_. You could however also use it when mixing two things together physically.  
_Röra ihop_ = same meaning as _blanda ihop_.

- _Blanda samman_ = _mix together_. When mixing two things together you usually use _blanda_. Though I don't think it's incorrect to say "röra samman" though.

And so on. There's also the noun _röra_ meaning either "a mess" or a food dish where things are mixed.

I don't envy anyone trying to learn this language!