Hun siger sandheden, selvom hun ser flov ud.

Because Danish normally uses the definite form in this expression: at sige sandheden = to tell the truth.

• sandhed = truth
• sandheden = the truth

So in this sentence, Danish is talking about the truth as a specific thing being told, not just the abstract idea of truth.

This is just an idiomatic difference between the languages.

In English, the natural phrase is tell the truth.
In Danish, sige sandheden is very common and natural.

Danish can also use fortælle sandheden, which is closer to English tell the truth, but sige sandheden is completely normal.

Selvom means although, even though, or though.

It introduces a clause that contrasts with the main idea:

• main clause: Hun siger sandheden
• contrasting clause: selvom hun ser flov ud

So the sentence structure is basically:

She tells the truth, although/even though she looks embarrassed.

Because selvom introduces a subordinate clause, and subordinate clauses in Danish do not use normal main-clause V2 word order.

In a main clause, Danish often puts the finite verb in second position:

• Hun ser flov ud.

But after a subordinating conjunction like selvom, the subject normally comes before the finite verb:

• selvom hun ser flov ud

So selvom ser hun flov ud would be wrong here.

There is a difference in meaning:

• hun er flov = she is embarrassed
• hun ser flov ud = she looks embarrassed

So ser ... ud describes appearance, not necessarily inner reality. The sentence is saying that she appears embarrassed, not directly stating that she actually is embarrassed.

Because se ud is a common Danish verb expression meaning to look / to appear.

In this pattern, the adjective often goes between ser and ud:

• ser træt ud = looks tired
• ser glad ud = looks happy
• ser flov ud = looks embarrassed

So the order is very natural:

• hun ser flov ud

Not:

• hun ser ud flov

Flov usually means something like:

• embarrassed
• ashamed
• awkward about oneself

It is not exactly the same as shy.
A shy person is generally timid or reserved, but a flov person often feels embarrassed because of a situation.

So hun ser flov ud suggests that she looks uncomfortable or embarrassed.

Because each clause needs its own subject.

The sentence has two clauses:

1. Hun siger sandheden
2. selvom hun ser flov ud

Even though both clauses refer to the same person, Danish still repeats hun. That is normal and required here.

Both are present tense forms:

• siger = present tense of sige
• ser = present tense of se

Danish verbs do not change according to person the way English verbs sometimes do. So the same present-tense form is used with different subjects:

• jeg siger
• du siger
• hun siger

and

• jeg ser
• du ser
• hun ser

Yes. That is completely correct.

If you move the selvom clause to the front, the main clause follows normal Danish inversion rules:

• Selvom hun ser flov ud, siger hun sandheden.

Notice that after the fronted subordinate clause, Danish puts the finite verb before the subject in the main clause:

• siger hun
• not hun siger

So this is a very useful pattern to learn.

Because selvom hun ser flov ud is a subordinate clause, and Danish often marks that with a comma.

However, there is an important detail: Danish has allowed two comma systems. In one system, you put a comma before subordinate clauses; in the other, you often do not.

So you may see both:

• Hun siger sandheden, selvom hun ser flov ud.
• Hun siger sandheden selvom hun ser flov ud.

The version with the comma is very common and completely correct.