  We show how a bijection due to Biane between involutions and labelled Motzkin
paths yields bijections between Motzkin paths and two families of restricted
involutions that are counted by Motzkin numbers, namely, involutions avoiding
4321 and 3412. As a consequence, we derive characterizations of Motzkin paths
corresponding to involutions avoiding either 4321 or 3412 together with any
pattern of length 3. Furthermore, we exploit the described bijection to study
some notable subsets of the set of restricted involutions, namely, fixed point
free and centrosymmetric restricted involutions.
