Should there be a balance between sinful acts and virtuous acts in life?

This sounds most like Nichiren Buddhism, which, being a pure land school, was able to acknowledge wrong doing and bad conduct by people that genuinely wanted enlightenment.

Of course, certain sutras and traditions like zen talk about breaking the precepts and an absence of goodness. And the tantric tradition welcomes sex. But suggesting we balance good with bad karma, is as much a misunderstanding as saying that zen encourages amoral behaviour.

I'm not saying it's just it's just rhetoric, but Chinese Buddhism built on top of early Buddhism (see different bodhidharma) -- and didn't just abandon it -- -- and early Buddhism didn't encourage sinful acts.

Both are fire, the goal is to abandon both good and evil. Below is a quote from Theory and Reality by Luang Pu Chah

So the Buddha examined the causes and conditions underlying existence and rebirth. As long as he had not yet fully penetrated the matter and understood the truth, he continued to probe deeper and deeper with a peaceful mind, reflecting on how all things, peaceful or not, come into existence. His investigation forged ahead until it was clear to him that everything that comes into existence is like a lump of red-hot iron.

The five categories of a being's experience (khandhas) are all a lump of red-hot iron. When a lump of iron is glowing red-hot, is there anywhere it can be touched without getting burnt? Is there anywhere at all that is cool? Try touching it on the top, the sides, or underneath. Is there a single spot that can be found that's cool? Impossible. This searing lump of iron is entirely red-hot. We can't even attach to serenity.

If we identify with that peace, assuming that there is someone who is calm and serene, this reinforces the sense that there is an independent self or soul. This sense of self is part of conventional reality. Thinking, "I'm peaceful", "I'm agitated", "I'm good", "I'm bad", "I'm happy", or "I'm unhappy", we are caught in more existence and birth. It's more suffering. If our happiness vanishes, then we're unhappy instead. When our sorrow vanishes, then we're happy again. Caught in this endless cycle, we revolve repeatedly through heaven and hell.

I think that Buddhism might see "virtuous" as "absence of sin" -- certainly monks' lives, for example, seem to me to emphasise avoiding various forms of harmful or potentially harmful behaviour.

And it isn't necessary to balance not-doing-harm with doing-harm. A reason for not doing harm (or for behaving virtuously, for being kind) is to have no remorse:

Skillful virtues have freedom from remorse as their purpose, Ananda, and freedom from remorse as their reward.

Trying to counter-balance that with harmful behaviour is missing the point.

It may be true that people are warned not to imagine (and not to aspire to) remaining in a permanent heaven. But what's meant to counter-balance that, to be an antidote, isn't being evil -- it's being (or becoming, or remaining) aware of the characteristics of existence -- e.g. anicca and anatta for non-grasping.

I think that's the Buddhist doctrine, and the kind of reason why jhana isn't the ultimate goal (nibbana is).

The limited extent to which the statement is true might be to do with conceit -- i.e. thinking, "I am a good person! Better than that other person!", might be some kind of hindrance. Perhaps a teacher might encourage someone not to grasp that kind of self-image; but doing that "by doing evil" seems to me quite possibly a mistaken way, and not a skilful way, to do that.

In Vajrayana school as presented by Chogyam Trungpa Rinpoche there's a prominent line of teaching about so called "spiritual materialism".

SM is when we use spirituality as a way to accumulate the imaginary dharmic wealth and use it to pump up one's ego. SM is not as much about what you do on the outside as it is about your hidden attitude and motives.

The thing with ego, its attitudes and motives are always well masqueraded. Ego is a master of disguise, it has endless capacity for self-rationalization. Even though its real motives are always selfish, it will infinitely pretend to everyone and to itself that it acts entirely in the interests of the common good and other highest ideals.

Naturally, stalking the ego's, exposing its tricks, leaving it without nutrition, and ultimately making it starve to death is a most important practice in Vajrayana. The student and teacher employ a wide variety of techniques to accomplish this extremely difficult but important aspect of Buddhist training.

On the student's end, one technique used to knock the ground from under the ego's feet (🤔 does ego have feet?) - is to deliberately perform whatever acts the ego finds shameful -- the opposite of its normal food. This includes performing what may look like the sinful acts. The difference between these balancing actions and the regular unskillful acts is huge, while the regular misdeeds are done under the influence of passion, aggression, or confusion - these special skillful downfalls are done with the cool head and clear mind, aiming at breaking the ego's pretense of decency and consistency.

How is this done in practice? Specific acts are entirely up to the student to improvise. They range from something as mild as quietly swearing(cursing) about someone or something in the privacy of one's room, to breaking one or more of the five precepts in a highly public and visible way. I heard stories of lineage teachers going as far as to order the student to go steal something from a shop or even to (!) shoot a deer. Needless to say, picking an appropriate ego-cidal act requires great sensitivity and a mature sense of judgment on the teacher's and student's part. This is one of the reasons that the real Vajrayana is almost always a secret teaching presented only to the most advanced and capable of the students.

In the long term the practice gets more and more refined, until even an overly strong balancing act may be seen as ego's attempt to pass e.g. for an advanced Vajrayana practitioner, something that now has to be balanced out by its opposite and so on until the entire structure of balancing and counterbalancing acts collapses along with the ego itself.

This death of ego is the greatest disappointment a human can encounter in their life. The rock bottom thus hit serves as an unshakable foundation for the ultimate kindness, sincerity, courage, sanity, sobriety, and authenticity known as the Buddhist Enlightenment.

No. Lord Buddha always encourages and praises those who strives aiming to eliminate all traces of unvirtuous action from their conduct and bring to consummation the virtuous actions.

“And what, monks, is right effort? (i) There is the case where a monk generates desire, endeavors, activates persistence, upholds & exerts his intent for the sake of the non-arising of evil, unskillful qualities that have not yet arisen. (ii) He generates desire, endeavors, activates persistence, upholds & exerts his intent for the sake of the abandoning of evil, unskillful qualities that have arisen. (iii) He generates desire, endeavors, activates persistence, upholds & exerts his intent for the sake of the arising of skillful qualities that have not yet arisen. (iv) He generates desire, endeavors, activates persistence, upholds & exerts his intent for the maintenance, non-confusion, increase, plenitude, development, & culmination of skillful qualities that have arisen. This, monks, is called right effort. (https://www.dhammatalks.org/suttas/SN/SN45_8.html)

“There is the case where a monk might say, ‘Although goodwill has been developed, pursued, handed the reins, taken as a basis, steadied, consolidated, and well-undertaken by me as my awareness-release, still ill will keeps overpowering my mind.’ He should be told, ‘Don’t say that. You shouldn’t speak in that way. Don’t misrepresent the Blessed One, for it’s not right to misrepresent the Blessed One, and the Blessed One wouldn’t say that. It’s impossible, there is no way that—when goodwill has been developed, pursued, handed the reins, taken as a basis, steadied, consolidated, and well-undertaken as an awareness-release—ill will would still keep overpowering the mind. That possibility doesn’t exist, for this is the escape from ill will: goodwill as an awareness-release.’ (https://www.dhammatalks.org/suttas/AN/AN6_13.html)

“Monks, even if bandits were to carve you up savagely, limb by limb, with a two-handled saw, he among you who let his heart get angered even at that would not be doing my bidding. Even then you should train yourselves: ‘Our minds will be unaffected and we will say no evil words. We will remain sympathetic, with a mind of goodwill, and with no inner hate. We will keep pervading these people with an awareness imbued with goodwill and, beginning with them, we will keep pervading the all-encompassing world with an awareness imbued with goodwill—abundant, enlarged, immeasurable, free from hostility, free from ill will.’ That’s how you should train yourselves. (https://www.dhammatalks.org/suttas/MN/MN21.html)