The first transformation is an Achmatowicz rearrangement, followed by protection and selective reduction; the last is a Johnson-Claisen rearrangement.
The acetal protects the other oxygenated position during hydride reduction. The orthoester transfers an acetate-derived two-carbon unit through a ketene acetal; it does not directly substitute the allylic OH with an intact acetate ion.
Figure 1.
Atom maps for the allylic ketene-acetal and ammonium-ylide rearrangements
. Details.
For a concrete choice, draw the intermediate in the planar ring orientation used in the question and put the allylic OH on the upper face, with a solid-wedge C-OH bond. Fix either configuration at the R-bearing and methoxy-bearing centres and retain it throughout; no absolute configuration is specified by the question.
Draw the six-membered rearrangement array with its allylic C-C-C segment along the same ring edge and the O-ketene-acetal tether above that edge. Suprafacial transfer forms the new ring-carbon–CHCOEt bond on that upper face: draw it as a solid wedge, replacing the original allylic OH with the shifted double bond. The R-bearing and methoxy-bearing stereocentres retain their configurations. The opposite OH epimer transfers the tether from the opposite face and gives the opposite new stereocentre. Favor the chair-like transition structure that puts the larger tether substituents outward. This explicitly correlates the selected OH face with the new bond rather than invoking a freely rotating allylic carbocation.
Propionic acid is a weak, relatively high-boiling acid: its normal boiling point is about , compared with about for acetic acid. It provides mild proton catalysis while remaining in the hot orthoester mixture more effectively. This is useful because ketene-acetal formation and the thermal Johnson-Claisen rearrangement require heating, whereas strong acid would threaten the pyran acetal.
This is a practical catalyst choice, not an absolute mechanistic prohibition on acetic acid; acetic-acid-catalysed orthoester rearrangements are also possible.

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