What I expected to happen:
Entities next to the invisible spider to render properly
What actually happened:
That entity didn't render properly
How to reproduce (Spider):
1. Spawn a spider
2. Throw splash potion of invisibility on the spider
3. Spawn any entity next to the spider
4. Notice how the entity looks weird
How to reproduce (Creeper):
1. Spawn a creeper
2. Make sure entity shadows are set to off in the video settings. (Options... -> Video Settings... -> Entity Shadows)
3. use
/entitydata @e[type=creeper,c=1] {powered:1b}
4. Spawn any entity next to the creeper
5. Notice how the entity looks weird
How to reproduce (Wither):
1. use
/summon wither ~ ~ ~ {Health:0.1f,NoAI:1b,ActiveEffects:[{Id:14,Amplifier:1b,Duration:10000}]}
2. Spawn any entity next to the wither
3. Notice how the entity looks weird
Probably invalid since bosses can no longer have effects without using commands
Notes:
If the spider/creeper dies, moves far enough, or is no longer invisible/charged,the entity starts rendering normally.
Other Entities can be seen through the bugged entity
This cannot be used to see entities through wall as shown in pictures 7 and 8.
The creeper one doesn't usually happen unless Video Settings: Entity Shadows is off, because shadows reset the depth mask.
Code analysis:
Spider: https://bugs.mojang.com/browse/MC-79697?focusedCommentId=290622&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel#comment-290622
Creeper: https://bugs.mojang.com/browse/MC-98888?focusedCommentId=296002&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel#comment-296002
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Confirmed.

Confirmed. Works with Cave Spiders. Does not work with any other entities except spiders. Effect:
Changes a portion of the mob's body transparent/translucent.

Confirmed for
15w40b

Confirmed for
16w02a

Please link to this comment in the description of the report.
The following is based on decompiled version of Minecraft 1.8 using MCP. All method and class names are the names used in the decompiled version.
The reason which this happens is very likely that the public void func_177148_a(EntitySpider p_177148_1_, float p_177148_2_, float p_177148_3_, float p_177148_4_, float p_177148_5_, float p_177148_6_, float p_177148_7_, float p_177148_8_)
method of the net.minecraft.client.renderer.entity.layers.LayerSpiderEyes
class disables the depth mask if the spider is invisible but is not enabling it again after it rendered the spider.

Probably relates to MC-9553
Confirmed for 16w15a and 16w15b
Confirm for 1.9.3 pre-2
Confirmed for 1.9.4
Confirmed for 16w20a

I accidentally posted a really small one, so I re-posted it bigger.

Look at the draw order of the creeper and the water!

That makes a lot of sense now!
I built a little arena some days ago and used command blocks to spawn the mobs. And charged creepers were included in the list. I noticed that sometimes mobs that spawned next to those creepers usually had their face texture inverted. My life is explainde now.

Entty shadows on allows me to replicate the Spider one, have to turn it off for the creeper(and makes this weird thing http://imgur.com/a/yFfb3)
Still happens with spiders and withers in 1.12.1, charged creepers are fixed.
Also affects creepers when entity shadows are off.

Confirmed for 1.12.2
Edit: I moved the creepers a block from the charged creepers. Now, one of them renders just fine but another one the same exact distance away but in the opposite direction renders incorrectly when I'm a certain distance away from it north (east/west doesn't seem to matter) (closer to the charged creeper)
I noticed this in my world I made for resource pack testing. I had some mobs in barrier cages, and creepers in some of them. I summoned a creeper, a creeper with no AI, a charged creeper, and a charged creeper with no AI in a row. The first issue I had was that the last charged creeper (with no AI) did not render properly. However, this went away as soon as I charged it and then left/rejoined the world. Then, I noticed the second creeper (not charged, with no AI) was not rendering correctly. I left/rejoined the world several times and switched resource packs several times, but could not fix it, so I searched on the issue tracker and found this. (Sorry if that was a little lengthy- second comment/whatever on the issue tracker ever...)
Confirmed for 1.13-pre8
Confirmed for 1.13
Confirmed for 18w30b