Learn Spanish (Latin America): Le até los cordones sin que él se volteara. - I tied his shoelaces without him turning around.

Because there is a change of subject. You are tying the shoelaces, but he is the one who isn't turning around. When the subject changes after sin, you must use sin que followed by a conjugated subjunctive verb. If you were doing both actions (like tying your own shoes without turning around), you would just use sin plus the infinitive: Me até los cordones sin voltearme.

Why does this sentence use se volteara instead of the present subjunctive se voltee?

The tense of the subjunctive verb needs to match the timeframe of the main action. Because the main verb até (I tied) is in the past tense, the negative condition introduced by sin que also has to be in the past. Therefore, we use the past (imperfect) subjunctive se volteara instead of the present.

Why does the sentence say Le até los cordones instead of Até sus cordones?

In Spanish, when an action affects someone's body parts or clothing, you usually use an indirect object pronoun (like le) to show who is affected, plus a definite article (like los) instead of a possessive adjective (sus). The literal structure is closer to 'I tied the shoelaces to him.'

I usually see darse la vuelta for turning around. Is voltearse used exactly the same way?