What is the converse of "hence"?

As I understand it, you're trying to show the fundamental connection between the first state and the second. The second state caused the first, so it should follow that its present when the first is. You're not looking for the "opposite", (an imprecise term usually meaning inverse) you want the converse. Using that understanding, we could have:

Its raining. Naturally, its cloudy.

Where its refers to the local weather condition by context, and the lack of change in context implies that the two are related, so naturally implies that there's a fundamental relation. You could also use:

Its raining. Of course it's cloudy.

Hinting that raining should inherently imply cloudy.

Its raining. Its obviously also cloudy.

Explicitly saying that there's a known relation there.

Alternatively to all of that, you could join the two sentences, resulting in the smoother:

Its raining, so naturally its cloudy.

Because would be the most direct opposite. Try a sentence where causality is clearer. The power is out, hence we can't watch the show tonight. We can't watch the show because we have no power.

Antonym-pairs can only exist within the existence with binary states.

If we are considering binary categorization of proximity, then the following would be an atonym-pair (if it's not here, it could only be there):

Here and there

What is the antonym of "Now"? It cannot have an antonym-pair, because it would be an antonym-triad

now, preceding, subsequent

If we constrain ourselves to a situation where time and space are the only two dimensions to consider, then the following would an antonym-pair in such a constricted environment

When and where

If we consider {here & there} as sources of causation, then the following would be antonym-pair.

Hence and thence

Remoteness and Proximity can form an antonym-pair.

Hence, "hence" describes causality due to proximity.
And thence, "thence" describes causality due to remoteness.

However, if we decide to form a binary state between time and space, then the following is an antonym-pair.

Hence and whence.

"Hence" denotes progression from a milestone. Where the milestone could be achievement of a concept or a location. Saying "hence" implies we can now move on consequent to what we have done in the past to achieve the milestone. Due to that causative relation of past to future, "hence" frequently implies "because".

Therefore, in this sense, we might wish to use progression back in time as the contrast of "hence"'s forward progression in time. For example,

Excessive human activity exacerbated global worming. Hence, we should reduce human activities.

To perform a reversal of causation indirection, we would say,

We should reduce human activities because excessive human activity exacerbated global warming.

Therefore, in this sense, the following are an antonym-pair.

Hence and because.

What about since and for?

There are a lot of clouds, hence it is raining.

Since there are a lot of clouds, it is raining.

It is raining, since there are a lot of clouds.

It is raining. For there are a lot of clouds.

I've seen hence and since forming a pair quite often. Furthermore, it is not at all unusual or frowned upon to begin a sentence with "For, ..." (even though I personally do not like it).

If hence is loosely used to mean in the future, then one converse could be erstwhile, which means former (and could loosely used to mean in the past).

http://dictionary.cambridge.org/dictionary/british/erstwhile?q=erstwhile

Instead of:

"Look to my coming on the first light of the fifth day. At dawn, look to the east!"

Gandalf the White could have said:

"Look to my coming at first light, five days hence. At dawn, look to the east!"

On the fifth day, sometime around 06:30, Merry* might have said:

"Remember Gandalf's instructions, five days erstwhile?"

* If he did, I'm fairly sure it wasn't recorded by Tolkien.