Snúðu þér til hægri og horfðu á mig.

Because snúa often appears as the reflexive phrase snúa sér, meaning turn oneself / turn around.

So:

• snúa = to turn something
• snúa sér = to turn oneself

In this sentence, the person is being told to turn their own body, so Icelandic uses the reflexive pronoun:

• Snúðu þér = Turn yourself / more naturally, Turn

Without þér, snúðu by itself would more naturally sound like turn it or would feel incomplete in this context.

Because the reflexive phrase snúa sér takes the dative case.

The reflexive pronoun changes by case:

• sig = accusative
• sér = dative
• sín = genitive

In the sentence, þér is the form used when speaking directly to one person, and it corresponds to the dative reflexive meaning yourself.

So the pattern is:

• snúa sér = to turn oneself

That is why Icelandic uses þér, not þig.

Snúðu is the singular imperative of snúa.

The imperative is the command form, so it is used for instructions like:

• Snúðu! = Turn!
• Komdu! = Come!
• Farðu! = Go!

In this sentence, the speaker is giving a command to one person, so the singular imperative is used.

Horfðu is also a singular imperative. It comes from horfa, meaning to look or to watch.

So:

• horfa = to look/watch
• horfðu = look!

The sentence contains two commands joined by og:

• Snúðu þér til hægri
• og horfðu á mig

So the full sentence is basically:

• Turn to the right and look at me.

That -ðu ending is very common in the singular imperative of many Icelandic verbs.

For example:

• snúa → snúðu
• horfa → horfðu
• koma → komdu
• fara → farðu

So if you are talking to one person and giving a command, this kind of ending is extremely common.

It is not the only imperative pattern in Icelandic, but it is a very frequent one and useful to recognize.

Til hægri means to the right.

This is a fixed directional expression in Icelandic:

• til hægri = to the right
• til vinstri = to the left

The preposition til normally takes the genitive case. In these expressions, it functions as part of an established phrase meaning direction or side.

For a learner, the best approach is to memorize til hægri as a whole expression rather than trying to translate each word too literally every time.

It is part of a phrase governed by til, which normally takes the genitive.

In practice, though, til hægri is best learned as a fixed phrase meaning to the right. Many learners find it easiest not to overanalyze this expression at first.

The important practical point is:

• til hægri = to the right
• til vinstri = to the left

Those are very common set phrases.

Because horfa normally takes the preposition á when it means look at.

So:

• horfa á einhvern = look at someone
• horfa á eitthvað = look at something / watch something

That is why Icelandic says:

• horfðu á mig = look at me

and not horfðu mig.

This is something English speakers often need to get used to: Icelandic verbs often require a particular preposition, and you usually have to learn the verb together with it.

Because mig is the accusative form of ég (I / me), and in this sentence á takes the accusative.

The relevant forms are:

• ég = I
• mig = me

So:

• á mig = at me

This is another good example of why learning Icelandic pronouns by case is important.

This is a very useful distinction.

• horfa á mig = look at me
• sjá mig = see me

Horfa á is an active action: you direct your eyes toward something.
Sjá is about perception: you notice or see something.

So in a command telling someone where to direct their gaze, horfðu á mig is the natural choice.

It is directed at one person in the normal singular imperative, so it is the usual direct way to give a command.

If you were speaking to more than one person, Icelandic would use plural forms, for example:

• Snúið ykkur til hægri og horfið á mig.

So the original sentence is singular:

• Snúðu þér ... horfðu ... = speaking to one person

Not especially. The sentence has a very straightforward command structure:

• Snúðu þér til hægri = Turn to the right
• og horfðu á mig = and look at me

This is a normal way to chain two imperatives together with og (and).

So the word order is natural and standard for spoken instructions.