Intuitionistic logic is not symmetric regarding truth and falsity.
There primary notion in intuitionistic logic is constructive truth and proofs not constructive falsity and refutation:
a statement is true if there is a proof for it.
And then we discuss various ways of constructions of proofs.
We don't discuss when a statement is false or how to construct a refutation.
We can easily build a dual logic of refutations in place of proofs.
This was noted in the literature by some students of L.E.J. Brouwer
(it has been years since I have looked at these,
I will add references if I recall them).
There are also further refinements of intuitionistic logic like linear logic which restore symmetry between truth and falsity.
Intuitionist negation is constructive but pay attention to the meaning of the intuitionistic negation.
It is not the same negation as classical one.
Asserting the statement $\lnot p$ means that
we have a construction that transforms any proof of $p$ to a proof of $\not$ (absurdity).
If $\bot$ has a proof then every statement has a proof by the $\bot$ rule
(there is a restriction of intuitionistic logic that treats $\bot$
simply as an arbitrary atomic proposition called minimal logic).
So $\lnot p$ does not meant that $p$ is false,
it means that it can never be true.
To understand the distinction we should think in the constructive terms not platonic terms.
$p$ not true means we have not constructed a proof of $p$.
$\lnot p$ means no one can ever construct a proof of $p$ and
we have a construction showing that.
In constructive mathematics
(in the general sense, not in the sense of constructive theories)
a proof is not a formal object according
to some system of fixed rules.
It is a construction.
It is a primary notion that
need not be further explained in terms of sets or linguistic objects
(though there the meaning of logical symbols are closely related to
the rules of their manipulation by design of those rules).
The rules of proof construction are not limited to
what is considered to be acceptable at the moment.
We have to consider the possibility that
tomorrow someone may come up with a new kind of constructing proofs.
The intuitionist negation is constructive
because it is shown by constructing a particular kind of proof,
it is not shown simply by nonexistence of proofs or objects.
To show that $\lnot p$ holds it is not enough that you show there are no proofs for $p$,
you have to give a construction that shows that there cannot a proof for $p$ ever.