I posted a request to mathgroup asking for guidance on why Solve[] and its brethren failed on the puzzle of the opposing clock hands. Two people have already responded with working versions from which we can draw the following conclusions,

1. We have to unwind the linedupedness[], unitTimeToHourAngle[], and unitTimeToMinuteAngle[] functions, and

2. We need to restrict the domain of the solution to the range from 0 to 1.

When we do this, we go from

In[]:= Reduce[unstraightness[unitTime] == 0 && 0 < unitTime < 1]
to
In[]:= Solve[Abs[Abs[4320 Mod[unitTime, 1/12] - unitTime*360] - 180] == 0 &&
0 < unitTime < 1, unitTime]

which does indeed produce the right answer. There are other ways to get there too; unwinding and restricting the solution to Reals or Rationals will work, though we get some redundant solutions that way.

Wolfram tech support had an even nicer solution, pointing out that my wrapper function (unstraightness[]) could take any kind of input, while the internal functions (unitTimeToHourAngle[] and unitTimeToMinuteAngle[]) are restricted to real numbers. This means that when these functions are introduced into the Reduce function, unitTime is not defined as a number, so the function is never evaluated. The simplest solution is just to change the definition of the internal functions to

In[]:= minuteAngle[unitTime_] := 4320 Mod[unitTime, 1/12]
and
In[]:= hourAngle[unitTime_] := unitTime*360 // N

whereupon

In[]:=Reduce[Abs[minuteAngle[unitTime] - hourAngle[unitTime]] == 0 &&
0 < unitTime < 1]

works like a charm.