My thinking is with respect to very serious issues wherein getting the correct resolution is very important. Usually the burden seems to be on one party or another, but I wonder if there are cases or situations where both parties are burdened with justifying their positions?
Are there cases or situations where both parties are burdened with justifying their positions?
Yes, but different standards are applied.
One example is the "reverse burden of proof" placed on the defendant in section 1(1) of the Prevention of Crime Act 1953:
Any person who without lawful authority or reasonable excuse, the proof whereof shall lie on him, has with him in any public place any offensive weapon shall be guilty of an offence...
The prosecution have to prove "beyond a reasonable doubt" that the article is actually an offensive weapon and that the named defendant possessed it on a particular date and that s/he was in a public place at the time.
Whereas the standard of proof for the defendant to show they had lawful authority or a reasonable excuse for having is the, lower, "balance of probabilities".
Similar legislation, for a similar offence, at section 139 of the Criminal Justice Act 1988 states that:
(4)It shall be a defence for a person charged with an offence under this section to prove that he had good reason or lawful authority for having the article with him in a public place.
(5)Without prejudice to the generality of subsection (4) above, it shall be a defence for a person charged with an offence under this section to prove that he had the article with him—
(a)for use at work;
(b)for religious reasons; or
(c)as part of any national costume.
Which, again, is on the balance of probabilities. See, for example, R v Lambert  UKHL 37:
If [a legal burden of proof on an accused] is created the matter in question must be taken as proved against the accused unless he satisfies the jury on a balance of probabilities to the contrary...
Generally speaking, the ultimate burden of proof is placed upon one party or another. But sometimes there is what is called a "burden of production" that involves some evidentiary showing that an issue is present before the burden of proof is implicated.
For example, no evidence needs to be presented regarding self-defense in a criminal case until the defense meets a burden of production to show that self-defense is an issue presented in a case, even if the ultimate burden of proof once the burden of production is satisfied and the issue goes to a jury is on the prosecution.