The answer to your question is, for the most part, no: Opposites do not attract.
Like-minded people attract; that is, the relationships they form are fuller and longer-lasting
than when paired with an opposite-minded partner. This is despite the fact that people often claim,
in surveys and when asked directly by interviewers, that they would like someone with personality
characteristics different from their own; revealed preference and lived experience show that they
really don't know what they want.
The article, Do People Know What They Want: A Similar or Complementary
Partner?, by a certain Pieternel Dijkstra,
reviews the hypotheses, and the research, better than I could give myself:
With regard to  “relative” mate preferences two hypotheses have been presented. First, according
to the “similarity-attraction hypothesis” individuals feel most attracted to potential partners who,
in important domains, are similar to themselves (e.g., Lucas, Wendorf, and Imamoglu, 2004). Similar
individuals are assumed to be attractive because they validate our beliefs about the world and
ourselves and reduce the risk of conflicts (e.g., Morry and Gaines, 2005). Not surprisingly
therefore, similarity between partners contributes to relationship satisfaction (e.g., Lutz-Zois,
Bradley, Mihalik, and Moorman-Eavers, 2006). Because a happy and long-lasting intimate relationship
contributes to both psychological and physical health (e.g., Berkman and Syme, 1994), similarity
between partners increases their own and their offspring’s chances of survival by helping maintain
(the quality of) the pair bond.
In contrast, according to the “complementarity hypothesis” individuals feel most attracted to
potential partners who complement them, an assumption that reflects the saying that “opposites
attract” (e.g., Antill, 1983). Complementary individuals are assumed to be so attractive because
they enhance the likelihood that one’s needs will be gratified (e.g., De Raad and Doddema-Winsemius,
1992). For example, young women who lack economic resources may feel attracted to older men who
have acquired economic resources and therefore may be good providers (Eagly and Wood, 1999). In
addition, from an evolutionary perspective, one might argue that seeking a complementary mate,
rather than a similar one, may help prevent inbreeding.
Ultimately, this is the punchline (emphasis mine):
Studies on mate selection have consistently found support for the “similarity attraction”
hypothesis. Homogamy has been reported for numerous characteristics such as physical attractiveness
(e.g., White, 1980), attachment style (e.g., Klohnen and Luo, 2003), political and religious
attitudes (e.g., Luo and Klohnen, 2005), socio-economic background, level of education and IQ (e.g.,
Bouchard and McGue, 1981). In contrast, support for the “complementarity hypothesis” is much
scarcer. Although many individuals occasionally feel attracted to “opposites”, attractions between
opposites often do not develop into serious intimate relationships and, when they do, these
relationships often end prematurely (Felmlee, 2001).
There is a particular area where people often do seek partners different from themselves, that's hot in the field of evolutionary psychology, and is extensively researched — that would the major
histocompatability complex, or MHC, in human mate choice. The MHC is a genomic region which codes for
protein receptors, MHC proteins, which are heavily involved in the immune system and autoimmunity;
the body uses them as antigens so that T cells and NK-cells, our immune's system "policemen", can
recognize foreign elements and differentiate them from "self". To protect against the great
diversity of bacterial and viral invaders, the MHC genome region needs to be highly polymorphic;
that means inbreeding at those loci (by mating with a "more-alike" partner) would be deleterious and
evolutionarily disfavored because inbreeding homogenizes alleles, which in turns means that mating with an "alike" individual would decrease the recognition ability of their offspring to detect pathogens. Thus, research often finds that humans often favor
dissimilar MHC alleles in their mates, which some researchers have hypothesized is mediated through
olfaction. Here is the abstract of one such study finding these preferences:
Preferences for mates that possess genes dissimilar to one's own at the major histocompatibility
complex (MHC), a polymorphic group of loci associated with the immune system, have been found in
mice, birds, fish, and humans. These preferences may help individuals choose genetically compatible
mates and may adaptively function to prevent inbreeding or to increase heterozy-gosity and thereby
immunocompetence of offspring. MHC-dissimilar mate preferences may influence the psychology of
sexual attraction. We investigated whether MHC similarity among romantically involved couples (N =
48) predicted aspects of their sexual relationship. All women in our sample normally ovulated, and
alleles at three MHC loci were typed for each person. As the proportion of MHC alleles couples
shared increased, women's sexual responsivity to their partners decreased, their number of extrapair
sexual partners increased, and their attraction to men other than their primary partners increased,
particularly during the fertile phase of their cycles.
Other than that, I can't think of anything else; the rule is pretty much, like attracts like.