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In 部, the right side radical is called the large village radical.  
For 陪, the left side radical is called the small village radical.

Why are their names different on different sides even though both are 阝?
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question
kanji etymology
The 阝-radical (or 部首?) in 部 and 陪
Perhaps not an answer, but the question is based on a false presumption that
"even though both are 阝". While the radicals may visibly look similar, the
kanji that they represent are quite different. For "big village", the
character is 邑, while for "small village", the character is 阜.

As radicals, "big village" is properly written as ⾢ (U+2FA2) or ⻏ (U+2ECF).
For "small village", it is ⾩ (U+2FA9) or ⻖ (U+2ED6). Notice that ⻏ (U+2ECF)
and ⻖ (U+2ED6) are two entirely different characters. In Unicode, the first is
called city while the second is mound radicals.
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answer
They're both written as 阝, but they're simplified representations of two
different kanji:

  * 邑 (7 strokes) is the full form of おおざと ("large village" おお＋さと）
  * 阜 (8 strokes) is the full form of こざと ("small village" こ＋さと）

As you say, 邑 always appears on the right, while 阜 always appears on the left.
Often, 阜 as 阝 is referred to as 阜偏（こざとへん)--the へん at the end tells you it's in
the left-side position. Likewise, 邑 as 阝 can be referred to as 邑旁（おおざとづくり),
where つくり indicates the right-side position.

The customary names of 部首 are somewhat arbitrary, and they don't always
communicate the meaning they lent to the kanji when they were formed, or if
they do, the connection to the meaning can be quite tenuous. Still, in the
case of 邑, it's called おおざと because it represents a (large) village. Let's
look at a couple examples of 邑 and see for ourselves:

  * 都 ("capital, metropolis") contains 邑 written as 阝, and in modern use it bears a clear semantic connection to the original meaning. 京都 is a large village indeed!
  * 郎, on the other hand, once was a place name meaning 良 good 阝(邑) village, but is now used as a simplification of 良 good 亻 person (man). Names like 太郎 certainly have no semantic connection to "big village"! 

So 邑 appears to represent large village, at least historically. 阜 on the other
hand has only a loose semantic connection to "small village". Etymologically
and in most kanji, it instead represents a hill or mound. It's a little more
apparent in the older form, which you can see on
[chineseetymology.org](http://www.chineseetymology.org/CharacterEtymology.aspx?submitButton1=Etymology&characterInput=%E9%98%9C).
It has three parts arranged vertically, representing a _terraced hill_. Why
then is it called こざと? By analogy to おおざと--the _opposite_ of an おおざと is a こざと,
and from appearances it is a 阝 written on the _opposite_ side.

Let's look at some examples for 阜:

  * The kanji pair 陰陽 (いんよう, often called yin and yang in English) contain 阜 as 阝 in its literal meaning of hill; the つくり of 陽 has a 日 sun with rays shining down onto a hill, creating a light side; 陰 is the other side of that hill, with the つくり representing its shadow. The semantic connection to "hill" is clear.
  * 陣 bears a relatively strong semantic connection to the 阜 it contains as 阝; it comes from 車 vehicles drawn up around a 阝 hill → army encampment → formation/position.

So we can see the "large village" and "hill, mound" meanings clearly, even if
they're sometimes obscured, and we've covered how "hill, mound" became "small
village". I believe this answers your question!

One last note: in modern Japanese, you're unlikely to see the characters 邑 and
阜 in their full forms unless you're using a 漢和辞典 or talking about kanji. I
developed a little mnemonic device to remember which one to turn to in my
dictionary (which is 縦書き, so this won't make sense if yours is 横書き and printed
left-to-right):

  * If 阝 is on the right, then it's on the right side of the dictionary (7 strokes, 邑)
  * If 阝 is on the left, then it's on the left side of the dictionary (8 strokes, 阜)

Note: in this entry I relied on Kenneth G. Henshall's _A Guide to Remembering
Japanese Characters_ as a primary source for character etymology.
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answer