What are good examples to show that, even if a regression of $Y$ on $X$ is heteroscedastic, a regression of $Y$ on a different independent variable $Z$ could be homoscedastic?
More formally, what are populations of triples $(Y_i, X_i, Z_i)$ for which all of the following hold:
a) $E[Y_i | X_i]$ is an increasing (or decreasing) function of $X_i$;
b) $Var[Y_i | X_i]$ is an increasing (or decreasing) function of $X_i$;
c) $E[Y_i | Z_i]$ is an increasing (or decreasing) function of $Z_i$;
d) $Var[Y_i | Z_i]$ is constant?
I am interested in examples where it is known or plausible that these properties apply to a population (ie they are not just quirks of particular samples that may not be representative). Although this scenario seems quite possible in theory, it is (to me at least) quite hard to find convincing examples.
Eventually I thought of the following. $Y_i$ is the number of years individual i is in full-time education. $X_i$ is i's average family income during the period of i's education. $Z_i$ is i's month of birth (1,...,12) in terms of the school year. Suppose the individuals i are in a country in which:
1) children are required to start school on reaching a certain birthday (so that they start at different times of the academic year, depending when their birthday falls);
2) compulsory education ends at the end of the academic year in which children reach a certain age, and most education beyond the compulsory period also ends at the end of the academic year;
3) children from families with higher incomes are more likely to stay in education beyond the compulsory period.
Then children from higher-income families are likely to spend longer in education on average, but the time they spend in education is likely to vary according to aptitudes and preferences, while children from lower-income families are more likely to spend in education only the compulsory period. Children born towards the beginning of the academic year are likely to spend longer in education on average, but there is unlikely to be much variation with month of birth in how much longer they spend (those born in different months being equally likely to be from higher-income families and to stay in education beyond the compulsory period).
Are there better (simpler, more obvious) examples?
Addendum: whuber has given an excellent theoretical answer showing how many examples can be generated. Answers involving specific examples with familiar variables (eg $Y$ = Income) would also be of interest.