The scales on the original Mannheim slide rule were just A [B, C] D.
Hemmi models 24, 34 and 34P (Frederick Post 1443 [1931-41]) used this scale
set.
Modern Mannheim slide rules add [S, L, T] scales to the back of the slide.
For an angle q on the S scale on the back of the
slide the value of sin q is in the same place on
the B scale on the front of the slide. The range of q is 34' to 90o and the corresponding
range of sin q is 0.01 to 1. For angle q on the T scale on the back of the slide the value
of tan q is in the same place on the C scale on
the front of the slide. The range of q is
5o 44' to 45o and the corresponding range of tan
q is 0.1 to 1. Note that sines are
read from the B scale and tangents are read from the C scale. Trigonometric
scales keyed this way are called "Mannheim" trig scales.
Here's the back of the slide from a Mannheim slide rule (Hemmi 50 from before
WWII). Note that the S scale starts at 34' (the first angle that is
actually labeled is 40') and the T scale starts at 5o 44'
(first labeled angle is 6o).
You may have noticed that one can find the sine of angles between 34'
and 5o 44' but not the tangent. For such angles you use
the fact that, to slide rule accuracy, tan q =
sin q.
Polyphase, Multiphase
Manufacturers used words like "Polyphase" or "Multiphase" to describe rules
with more than just the basic Mannheim scale set. Keuffel and Esser
called the scale set A [B, CI, C] D,
K and [S, L, T] "Mannheim Polyphase."
Rietz
K, A [B, C] D, L and [S, S&T, T] is the "Rietz" scale set.
It was most popular in Germany but I suspect all manufacturers made a least
one slide rule with Rietz scales. Hemmi Rietz rules have model numbers
in the range 60-79.
Rietz trigonometric scales are of interest. The S scale runs from
5o 44' to 90o (different from Mannheim rules), the
S&T scale runs from 34' to 5o 44' and the T scale runs
from 5o 44' to 45o (same as Mannheim). All three
scales are keyed to the C/D scales. For example, directly opposite
27o on the S scale is 0.454 on the C scale (sin 27o
= 0.454). Directly opposite 27o on the T scale is
0.510 on the C scale (tan 27o = 0.510). Directly opposite
3o on the S&T scale is 0.0523 on the C scale (sin 3o
= tan 3o = 0.0523. When using the S&T scale remember
to insert a zero between the number from the C scale and the decimal point--0.0523
not 0.523.) S scales that run from 5o 44' to 90o
and are keyed to the C/D scales are "Rietz S scales."
Hemmi Rules from WWII to 1955
From WWII to about 1955, Hemmi used Rietz S scales on their Mannheim
slide rules. At first glance the scales on the back of the slide look
like ordinary S and T scales. The T scale is ordinary; angles
run from 5o 44' to 45o andtan q ranges from 0.1 to 1 on the C scale. But the
angles on the S scale range from 5o 44' to 90o (not34' to 90o) and sin q ranges
from 0.1 to 1.0 on the C (not B) scale. There is no direct way to to
read the sine or tangent of angles smaller than 5o 44'.
I suppose there is some computational advantage to this arrangement but
it didn't prove popular and Hemmi reverted to "standard" Mannheim trig scales
about 1955. This experiment occurred on Hemmi models 30/32 (Post 1441),
34R, 34RK (Post 1444K, H-O 1762), 50 (Post 1452, H-O 1771), 50W
(Post 1452W), 50-20 (H-O 1772). A few Hemmi models (43, P43 and perhaps
others) started with a Rietz S scale and never changed.
Here is a picture of Hemmi hybrid trigonometric scale from a model 50 rule
made shortly after WWII. (The rule is marked "Made in occupied Japan.")
At first glance it appears to be an ordinary set of "Mannheim" trig scales
but note that the smallest labeled angle is 6o on both the S
and the T scales. On genuine Mannheim trig scales the smallest marked
angle on the S scale is 40'.
Inverted Trigonometric Scales
There is a theoretical computational advantage to inverted trig scales.
(TI scale runs from 45o at the left end to 5o 44' at
the right end; SI scale runs from 90o at left end to 5o
44' at the right end.) Hemmi used inverted trig scales on many of
its late model slide rules but the only Post models that use them are models
1445 and 1445P.
Folded Scales
In operations on the C and D scale; one frequently runs off scale.
For example, multiply 3 x 4 by putting the left end of the C scale opposite
3 on the D scale; move the cursor to 4 on the C scale and read the result
on the D scale. There isn't any D scale at this point! The result
is 12 but the D scale ends at 10. This problem can often be avoided if your
slide rule has folded scales. In the above example, when you realize
that the answer is off scale to the right, you can read the product (12) from
the DF scale--opposite 4 on the CF scale.
Off-scale problems are minimized if the folded scales are folded at the
center of theC/D scales, i.e., folded at square root of 10 = 3.16.
Hemmi made a lot of slide rules with scales folded at 3.16. Frederick
Post insisted that folded scales on their slide rules be folded at p (3.14). Folding at p
means the number under the hairline on the CF scale is p times the number on the C scale. Many engineering
calculations require multiplication or division by p
and p-folded scales provide an efficient way to
do so while still being good in handling off-scale results.
Darmstadt
If one does a lot of exponential calculations the scale set of choice is
"Darmstadt." The scales are: L, K, A [B, CI, C] D, COS, Sin, Tg and
[LL1, LL2, LL3]. The Sin scale is in degrees and gives the angle whose
sine is shown on the D scale. Similarly, the Tg scale gives the angle
whose tangent is shown on the D scale.
Hemmi made several Darmstadt rules (model numbers 130-139). There
were never any Frederick Post brand Darmstadt slide rules.
Here's the back of the slide from a Mannheim slide rule (Hemmi 50 from before
WWII). Note that the S scale starts at 34' (the first angle that is
actually labeled is 40') and the T scale starts at 5o 44'
(first labeled angle is 6o). 
Rietz trigonometric scales are of interest. The S scale runs from
5o 44' to 90o (different from Mannheim rules), the
S&T scale runs from 34' to 5o 44' and the T scale runs
from 5o 44' to 45o (same as Mannheim). All three
scales are keyed to the C/D scales. For example, directly opposite
27o on the S scale is 0.454 on the C scale (sin 27o
= 0.454). Directly opposite 27o on the T scale is
0.510 on the C scale (tan 27o = 0.510). Directly opposite
3o on the S&T scale is 0.0523 on the C scale (sin 3o
= tan 3o = 0.0523. When using the S&T scale remember
to insert a zero between the number from the C scale and the decimal point--0.0523
not 0.523.) S scales that run from 5o 44' to 90o
and are keyed to the C/D scales are "Rietz S scales." 
Here is a picture of Hemmi hybrid trigonometric scale from a model 50 rule
made shortly after WWII. (The rule is marked "Made in occupied Japan.")
At first glance it appears to be an ordinary set of "Mannheim" trig scales
but note that the smallest labeled angle is 6o on both the S
and the T scales. On genuine Mannheim trig scales the smallest marked
angle on the S scale is 40'.