Easy 1989
Designed by Minoru Abe circa 1989; this one purchased 2013.
(cardboard box and wood puzzle, 4.25" x 4.25" x 7/8")
The directions show 9 problems named A  C and 1  6 to remove the 2x1 keeper and rearrange the pieces
to read 1 2 3 4 oriented properly,
starting and ending with the blank piece in the turntable in its east position.
Any permutation / orientation can be achieved by a combination of cycling positions (1,3), (1,4), (2,4), (3,4)
while parking a piece in position (1,1),
and exchanging pieces located in positions (1,1) and (1,3) by using the turntable slot in its north position,
but finding a shortest sequence is harder.
We consider three rules for counting moves:
Rule 1:
A single move may pass through the turntable independent of initial orientation.
Rule 2:
A single separate move is required to use the turntable.
Rule 3:
Repositioning the turntable requires an additional move.
For example,
moving horizontally,
then rotating,
then moving vertically,
is 3 rectilinear moves if the turntable is east before the move,
or 4 rectilinear moves if it must first be rotated.
Although not the hardest,
Problem 3 is arguably the most asthenic,
and shown on the right above are 3 additional problems along those lines.
Here are the rectilinear moves for solutions using each of Rules 1, 2, and 3 for the total of 12 problems:
Problem A
40,
56,
67

Problem B
46,
64,
76

Problem C
58,
78,
92

Problem 1
42,
56,
67

Problem 2
52,
71,
84

Problem 3
26,
37,
46

Problem 4
60,
83,
99

Problem 5
57,
75,
88

Problem 6
53,
71,
83

Problem 7
32,
40,
48

Problem 8
38,
55,
68

Problem 9
22,
30,
35

A Sample Rectilinear Solution  Problem 3
Here a 26 rectilinear moves solution using Rule 1.
Solving using Rules 2 or 3 can work the same by inserting extra moves at the appropriate places
to achieve 37 or 46 rectilinear moves: