For instance, on Jun Da's table. 高 (tall) is listed at 134, while 矮 (short) is listed at 2027, 新 (new) is listed at 161, and 旧 (old) is listed at 915. It's worth noting that 老 (old) is listed at 179.
An exceptions to the rule includes 近 (near) and 远 (far) (374 and 386 respectively). You might also consider 大 (17) 小 (83) an exception; they are only 66 slots apart.
Is this because of a Chinese preference for the more positive quality, even though in theory, they should be "symmetrical?"
It's an interesting statistics.
I think the reason is that those words also have multiple other meanings.
For example, 高(134) vs 矮(2027),
高 could mean:
1. 由下到上距离大的，与“低”相对：～峰。～空。～踞。～原。～耸。～山流水（喻知己、知音或乐曲高妙）。～屋建瓴（形容居高临下的形势）。～瞻远瞩。
2. 高度：他身～一米八。
3. 等级在上的：～级。～考。
4. 在一般标准或平均程度之上：～质量。～消费。～价。～档。～手。～能物理。
5. 声音响亮：引吭～歌。
6. 敬辞，称别人的事物：～见。～就。～论。～寿。～堂。～徒。
7. 热烈、盛大：～昂。兴～采烈。
8. 显贵，道德水平高：崇～。清～。～风亮节。～尚。～雅。～洁。
9. 宗族中最在上之称：～祖。
10. 酸根或化合物中比标准酸根多含一个氧原子的：～锰酸钾。
11. 姓。
矮 could mean:
1. 人的身材短：～人。～矬。～个儿。～墩墩。
2. 高度小：～林。～墙。～屋。
3. 等级地位低：工资他比我～一级。
As you can see, 高 has much more meanings than 矮, 11 vs 3. So, it's no surprise to see 高 holds the position of 134, while 矮 just 2027.
In the examples given, the more frequent character has positive connotations, and the less frequent character, negative connotations. This is true of 高 (tall, 134), vs. 矮 (short,2027), 新 (new, 161), and 旧 (old, 915). Even 大 (big,17) and 小 (small,83).
One exception is 老 which means old in the sense of "aged" (179). That is considered a "good" quality in traditional Chinese society. Also, 近 (near) and 远 (far) (374 and 386 respectively) are both "neutral," with one no worse than the other.
Actually in the case of 高, it means high (high up), tall, and high level, while 矮 is just used to refer to short objects or people, the other opposite being 低. So 高 would refer to a tall person, a tall mountain, being high up, something being high level, etc.
Additionally 高 is used in more compound words, such as 高興, place names (高雄, 高山[literally and as a place name]) and 高 is also a family name, while 矮 isn't.
新 is similarly used in many place names (新加坡 - Singapore; also 新竹, 新北, etc.,) and it is in many compound words.
Also, both of those words being positive, they are frequently used in company names, brands, product names.
老, also being positive, is used in many business names as well as many compound words, very common are 老師, 老公, 老婆, 老闆, 老外, etc.
