Wiktionary says it’s literally “pig-dog” AKA bastard but I would imagine it has a more contextual meaning than that in German.
Is it a dog like a pig, or pig like a dog?
I thought the literal meaning was a dog used for hunting (or herding?) pigs.
Generally in German composite nouns the first part specifies the second - a noun1noun2 is a type of noun2 described by noun1 (and takes the gender from noun2). So a schweinehund is a hund (and takes the male gender from hund rather than the neuter gender from schwein).
Wiki says a type of dog used to hunt wild boars (Sauhund - this word is also encountered as an insult) used to also be called Schweinehund - but I have never encountered this usage (wild boars are hunted with firearms nowadays).
The usage in German is:
Schweinehund (noun, male): strong insult, referring to a man of reprehensible behaviour.
innerer Schweinehund (noun, male - literally inner pigdog): a relatively mild expression, metaphorically referring to weakness of will or courage in oneself or another person. Example: I had to overcome my inner pigdog every morning last week to make my morning run.
Why would one’s valuable hunting dogs be held in contempt so?
Hunting wild boars is quite dangerous; it’s not unheard of for one or more of the dogs to be injured or killed in the hunt.
So you don’t use your valuable, trained hunting dogs for this; rather it’s unproven young dogs or older but unreliable ones (“expendable”) that are used as schweinehunds.
While dogs are used to hunt boar nowadays, it’s very different from the way it was done back when the weaponry consisted of spears: now they’re not supposed to fight the pig, back then they were. So it’s not even necessarily the same kind of dog; being dumb enough to assault several hundred kilos of pissed-off tusked muscle isn’t a plus any more.
Grammatically, this is known as noun buildup or noun stacking and is a feature of some (all?) Germanic languages. English, as a fellow West Germanic language and sister to German, does the same thing, except it puts spaces between the constituent words, so “Schweinhund” is more or less the same thing as “pig dog”. If you want to use English words that are cognate to the German ones, you could say “swine hound”.
No odder than Dachshund - Badger dog.
We don’t have any Dachs here, so mine are Shizenhunds (spelling?), thanks to the cat. 
We have a friend in Germany that sends us Westerwalder Kuhe hunde’s. Cow dogs from Westerwald. They are cattle herding dogs.
It’s a pig dog. A pig herding dog. Like this.
My mother came from Germany and said it meant pigs rear!
The plural of “Hund” is “Hunde”* for starters, not “Hunds”. Also, if you’re trying to say what I think you are, the word is Scheiße** (or Scheisse), not Shize. It’s a hard S too, there’s no z sound in there.
Finally, in my experience all dogs are Scheißehunde 
- well, technically it depends on the grammatical role played by the dogs in the sentence. But word definitions are typically given using the nominative, so.
** Which, interestingly, has apparently no plural. I guess Germans can’t give two shits :).
Meta-Question: Should Insult Origin Threads belong in the BBQ Pit?
Boarhound.
In the US dogs are used to track and ‘bay up’ feral pigs, which are a significant problem, tearing up cropland and forest. They are indeed dangerous, and courage is an essential in what we call hog dogs out here. In the South they developed breeds for this, which double as wild cattle gatherers – it’s an actual occupation here, retrieving cattle that have gone feral, as well as clearing out wild pig infestations. Catahoula Cow Hog Dogs are the most common breed, part of a group of breeds called Cur Dogs (no slur intended, these are carefully bred dogs).
Just thinking of “Major” the Pig Dog from “Footrot Flats.” A dog you do not want to mess with!
Not necessarily - you get sheepdogs, foxhounds, fishermen, footballs, arseholes, seagulls etc etc in English.
Do you have some basis for this? It’s sounds a little “out of your ass”, or as we say in German “assengroppengehagen”.
Actually, English moves in a progression from separate words/phrases to abbreviated or hyphenated words to a compound word.
Thus to the morrow becomes to morrow then becomes to-morrow then tomorrow.
Or electronic mail becomes electronic-mail becomes e-mail becomes email.
And sometimes words stop in the process, and stay hyphenated. Following the rule that there are no absolute rules in English!
However, we seem to start with multiple word noun buildup. The famous “cat memes” on teh internets haven’t gotten to the point where we call them “catmemes”, but the day might come.