sich wälzen
to roll around
To roll or turn one's body back and forth on a surface. This can happen because of restlessness, pain, or pleasure.
Das Pferd wälzte sich genüsslich im Sand.
The horse rolled around contentedly in the sand.

to roll / ponder / shift
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to roll around
To roll or turn one's body back and forth on a surface. This can happen because of restlessness, pain, or pleasure.
Das Pferd wälzte sich genüsslich im Sand.
The horse rolled around contentedly in the sand.

The basic idea of 'wälzen' is heavy, slow turning or rolling. This image connects all the meanings: the rolling of a body, of a tree trunk, or the mental 'rolling around' of problems.
Er wälzte den schweren Stein mühsam zur Seite.
He laboriously rolled the heavy stone to the side.
In the meaning 'to roll back and forth', the verb is always reflexive. The reflexive pronoun (sich, mich, dich, etc.) is a fixed part of the expression here.
Ich habe mich die ganze Nacht unruhig im Bett gewälzt.
I tossed and turned restlessly in bed all night.
In the figurative sense of thinking, 'wälzen' is almost always used with abstract nouns such as 'Probleme' (problems), 'Gedanken' (thoughts), or 'Sorgen' (worries) as the object.
Statt zu schlafen, wälzte sie stundenlang ihre Probleme.
Instead of sleeping, she pondered her problems for hours.
When responsibility or costs are transferred to someone else, the structure is 'etwas auf jemanden (Akkusativ) wälzen' (to shift something onto someone). The preposition 'auf' with the accusative is crucial here.
Der Chef wälzt die unangenehme Aufgabe auf seine Mitarbeiter ab.
The boss shifts the unpleasant task onto his employees.
The expression 'Bücher wälzen' (to pore over books) is colloquial and is often used with a slightly ironic tone. It implies that one has to study very thick, heavy, or dry specialist literature.
Für die Prüfung musste ich das ganze Wochenende dicke Bücher wälzen.
For the exam, I had to pore over thick books all weekend.
to transmit / transfer
'übertragen' and 'wälzen' can both mean that responsibility, blame, or costs end up with another person. 'übertragen' is more neutral: a task or responsibility is passed on officially or matter-of-factly. 'wälzen auf', by contrast, sounds negative. It means that someone pushes something unpleasant onto others, often unfairly. The words can be confused because in both cases something moves from one person to another. If the transfer is normal or planned, 'übertragen' fits. If it feels like passing something off onto someone else, 'wälzen auf' fits.

Die Firma wälzt die zusätzlichen Kosten auf die Kunden ab.
The company shifts the additional costs onto its customers.
Vor dem Termin überträgt der Chef die Aufgabe an seine Kollegin.
Before the appointment, the boss transfers the task to his colleague.