Though of course prices and quantities depend on each other, the two ways of
representing the relationship are far from identical. I think it useful to
point out the following remark by Axel Leijonhufvud (Market adjustment
processes, Elgar Companion to Alfred Marshall, pp. 226-7):

Nothing better illustrates our confusions over neoclassical economics than
the universal habit of drawing Walrasian schedules in Marshallian space.
...
Supply-price and demand-price schedules are not loci of optimal points. A
supply-price is the minimum price a producer would accept in order to
continue producing the corresponding quantity. Any higher price would spell
abnormal profit. Similarly, a demand-price is the maximum price the consumer
will be willing to pay. Any price lower than pd(q) will obviously be
preferred. It would imply consumer¹s surplus.
The conceptual experiments underlying ps(q) and pd(q) schedules are thus
quite different from those generating the usual qs(p) and qd(p) functions.
Marshall¹s schedules are upper and lower boundaries of sets. Consequently,
it is in general not legitimate to treat Œquantity-into-price¹ functions as
inverses of Œprice-into-quantity functions¹ ... as has frequently been done
in the textbook literature.

Tiziano Raffaelli