People are puzzled at the tautology (p→q)∨(q→p).
I develop here two plausible interpretations of “(p→q)∨(q→p)” which are sometimes false.
To what end would you ever assert ‘p∨q’?
Why not instead find out which is true and merely assert that instead?
The main reason is that there are unspoken variables in both ‘p’ and ‘q’.
Here is a practical use of ∨:

A child is either a boy or a girl.

More formally, for every child x, (x is a boy)∨(x is a girl).
Here the disjunction serves a real and useful purpose.
It becomes useful in context of universal quantifiers.
Similarly p→q is useful in such contexts.
“All men are mortal.” is formally ∀x(man(x) → mortal(x)).

The sentential calculus, from the perspective of predicate calculus lacks quantifiers.
When a theorem from sentential calculus is imported into predicate calculus it is necessary to ‘close’ it by adding enough universal quantifiers to the whole theorem to eliminate all the free individual variables.
This is a convention that is introduced to formal logic well into the development of the subject.
It is a good and useful convention, sometimes left unsaid in even mathematical contexts; but it is only a convention.
The man on the street cannot be expected to know this and is justified in unconsciously interpreting (p→q)∨(q→p) as (∀x(p→q))∨(∀x(q→p)) rather than the mathematically conventional ∀x((p→q)∨(q→p)).
The latter is true but the former is not generally true.
This misinterpretation is especially likely if the universal quantifier notation is not known.

A closely related form of notational confusion is to confuse the proposition with the predicate and thus to see p→q as a relation between two predicates indeed embodied as sets.
In such a frame of mind p→q is taken to mean what is mathematically expressed as P⊆Q where P is {x|p} and p is imagined to somehow name x.
(p→q)∨(q→p) thus seems to mean (P⊆Q)∨(Q⊆P) which is not generally so.

In short people just don’t know the notation which is not surprising psychologically or philosophically.

### Intuitionist Math

Curiously the intuitionist school of mathematics says that they would never assert x∨y unless they could (at least in principle) either assert x or assert y.
This is so only when there are no hidden variables.
They would say that ‘j is even or j is odd’ if this was taken to mean
∀j(Integer(j) → (even(j) ∨ odd(j))).
For instance they will not even assert x∨~x unless they knew (or could know) which is the case.
I think that they would assert x∨~x when x = (2^{21000000}+1 is prime).
They would argue, I think, that it was certain that either one or the other could be asserted.
They would object for x = (axiom of choice).
### Material Implication

A more general objection heard from newcomers to logic is that p→q is to be considered true when p is false.
I think that this reluctance stems from explaining “p→q” as meaning “p implies q”.
Implication has connotations of an explanation of how it is that q must be true if p is.
I recall this misgiving as I began to study logic.
The misgiving was dispelled when someone pointed out that for math, the meaning of p→q could be entirely captured as ~p∨q.
I would be curious about the outcome of the Wason test if the purported rule were given in the ~p ∨ q form.
This does not preclude the possibility of explanations, merely that p→q is a lesser claim and all that math needs.