Here's a musing. Take the Calvinist acronym TULIP (Total Depravity, Unconditional Election, Limited Atonement, Irresistible Grace, Perseverance of the Saints). I've heard some people of the Reformed persuastion claiming that these five points stand or fall together, and that just seems false to me, so I did some pointless speculation to think through some possible combinations short of the five points.
You can believe in total depravity without believing in any atonement, grace, or salvation. This position might be called universal damnationism. It's a perfectly consistent view, but it doesn't involve any election, atonement, grace, or perseverance. This is T without any of the others. (I suppose you might technically say that there aren't any conditions on election, since there's no election, there are limits to the atonement -- limiting it to no one, all the grace God shows is irresistible, but he shows it to no one, and all the saints, which is none of them, persevere -- but this is a pyrrhic victory).
I think the universalist position can be consistent also, and that involves TUIP. My arguments against this view are from scripture, not consistency. (Of course most people who say they are TUIP, I believe, really do hold to L but don't realize it, since I think the potential atonement view is consistent with limited atonement, and limited atonement just says that only some will actually be saved, but this is a controversial view that I won't insist on to make this point. See this and this other thing for more on why I think that.)
I think it's possible to hold to P without holding to U, L, or I (and it's debatable whether this view holds T in the same way Calvinists do). This is a moderate Arminianism common among the SBC. On this view, God's work is necessary for people to repent, but they can resist God, and so God's sovereignty in salvation is incomplete. There's no real election, the atonement is potentially available to all in the most extreme sense, and God's grace can be resisted. However, God will sustain those who do end up repenting, so in a way the all-important Arminian freedom to choose salvation is no longer available to those saved in order to stop believing. That's a strange consequence, but it's a consistent view. So this view has either TP or just P without the rest of them (depending on whether you think this T is worth the name, which I think it could be if God give the disposition to be able to believe as something necessary for salvation, and those who don't have it still can't repent). (Of course, I should also say that I think this view DOES hold to limited atonement, since I think everyone does in one sense except universalists. Then it would be TLP or LP.)
Now I think there are even stranger possibilities. There's sort of a reverse of the previous possibility. The Calvinist is right about our condition before repentance. We're totally depraved, needing a complete work of God to repent, something God doesn't base on anything we've done to try to deserve it, and God has chosen to limit the atonement to those who will repent and persevere. We can't resist this grace. So we have TULI. However, on this view, God gives us Arminian free will to fall away once we're saved, so there's a denial of P. We can resist the continuance of grace, anyway. Maybe you'll say this is denying I. If so, then it's TUL without the others. The election in this case isn't election to permanent salvation but election to potential salvation, something that can be lost. You might even think that's not worthy of U. If so, then it's still TL. I don't think you could drop either of those. I think this is a consistent view, though I don't know if anyone believes it.
There's another really strange view that some people think Calvinists hold. People choose to follow God, and others don't. Of course, little did we know but God has chosen those who will be saved and those who won't, and it's totally independent of what anyone has done or will do. It's independent of trusting in Christ, of good works, of even knowing about God. It's just arbitrary. This election is unconsiditional, therefore, in an extreme way. The atonement is also limited. Since believing is independent of God's work, this view denies T. The grace that saves is independent of any resisting, so I'm not sure if you would say the view accepts I. P has the same problem, and I would imagine you would resolve it the same way as I. You might think of the view as ULIP if you think of the irrestibility as an inability to prevent salvation, and perseverance has the same sense. If you think of irresistibility as resisting actual belief and perseverance in a similar way, then it would be UL.
Here's another view. God saves people if they're left-handed (or pick any arbitrary criterion). This denies unconditional election. Otherwise, all the Calvinist theses are maintained, so we have TLIP.
So here are the possibilities I've listed:
TP or P or TLP or LP
TULI or TUL or TL
ULIP or UL
These are all consistent positions as far as I can tell, and the issue is which set of letters best describes each view, not whether the view is consistent. Therefore, I claim that there are at least seven possible consistent combinations of views involving these letters. I would guess that there are more combinations also, but I don't want to try all of them.