one-sided pentominoes:                  Torsten Sillke, May 1993
                                update 1999-01-15: the odd prime L5 rectangles
                                update 1999-03-20: the odd prime Y5 rectangles
                                update 1999-04-14: the odd prime P5 rectangles

There are three pentominoes, which tile rectangles and are handed.
These are: L, P, and Y.

      prime rectangles                   odd       who and when
    +----------------------------------+-----+------------------------------
 L5 | 2x5, 13x55, 15x39, 17x35, 19x25  | yes | Michael Reid [until may 1998]
 Y5 | 5x10                             |  no | Yuri Aksyonov 1999-03-20
 P5 | 2x5                              |  no | Michael Reid 


one-sided tetrominoes:
The handed L4 is long known to have odd rectangles.
(D. A. Klarner, AMM 70:7 (Sep. 1963) 760-61 E1543).

The L5 rectangles:
The smallest odd rectangle with right handed L5 is 19x25:

+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ 
|  |           |  |           |  |           |     |     | 
+  +--+--+--+  +  +--+--+--+  +  +--+--+--+  +  +--+  +--+ 
|           |  |           |  |           |  |  |  |  |  | 
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+  +  +  +  + 
|  |           |  |           |  |           |  |  |  |  | 
+  +--+--+--+  +  +--+--+--+  +  +--+--+--+  +  +  +  +  + 
|           |  |           |  |           |  |  |  |  |  | 
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+  +--+  + 
|  |           |  |           |  |           |     |     | 
+  +--+--+--+  +  +--+--+--+  +  +--+--+--+  +--+--+--+--+ 
|           |  |           |  |           |  |           | 
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+  + 
|     |     |     |     |  |           |  |           |  | 
+  +--+  +--+  +--+  +--+  +--+--+--+  +  +--+--+--+  +--+ 
|  |  |  |  |  |  |  |  |           |  |           |  |  | 
+  +  +  +  +  +  +  +  +--+--+--+--+--+--+--+--+--+--+  + 
|  |  |  |  |  |  |  |           |  |     |           |  | 
+  +  +  +  +  +  +  +--+--+--+--+  +  +--+--+--+--+  +  + 
|  |  |  |  |  |  |  |           |  |  |           |  |  | 
+--+  +--+  +--+  +--+--+--+--+  +  +  +--+--+--+  +--+  + 
|     |     |     |           |  |  |  |     |  |  |     | 
+--+--+--+--+--+--+--+--+--+  +--+  +  +  +--+  +--+--+--+ 
|  |           |     |     |  |     |  |  |  |           | 
+  +--+--+--+  +  +--+  +--+--+--+--+--+  +  +--+--+--+--+ 
|           |  |  |  |  |  |           |  |           |  | 
+--+--+--+--+--+  +  +  +  +--+--+--+  +  +--+--+--+--+  + 
|  |           |  |  |  |           |  |  |           |  | 
+  +--+--+--+  +  +  +  +--+--+--+--+--+--+--+--+--+  +  + 
|           |  |  |  |  |           |  |           |  |  | 
+--+--+--+--+--+--+  +--+--+--+--+  +  +--+--+--+  +--+  + 
|  |           |     |           |  |           |  |     | 
+  +--+--+--+  +--+--+--+--+--+  +--+--+--+--+--+--+--+--+ 
|           |  |  |           |  |     |     |     |     | 
+--+--+--+--+--+  +--+--+--+  +--+  +--+  +--+  +--+  +--+ 
|  |           |           |  |  |  |  |  |  |  |  |  |  | 
+  +--+--+--+  +--+--+--+--+--+  +  +  +  +  +  +  +  +  + 
|           |  |  |           |  |  |  |  |  |  |  |  |  | 
+--+--+--+--+--+  +--+--+--+  +  +  +  +  +  +  +  +  +  + 
|  |           |           |  |  |  |  |  |  |  |  |  |  | 
+  +--+--+--+  +--+--+--+--+--+  +--+  +--+  +--+  +--+  + 
|           |  |     |     |     |     |     |     |     | 
+--+--+--+--+--+  +--+  +--+--+--+--+--+--+--+--+--+--+--+ 
|  |           |  |  |  |  |  |           |  |           | 
+  +--+--+--+  +  +  +  +  +  +--+--+--+  +  +--+--+--+  + 
|           |  |  |  |  |  |           |  |           |  | 
+--+--+--+--+--+  +  +  +  +--+--+--+--+--+--+--+--+--+--+ 
|  |           |  |  |  |  |  |           |  |           | 
+  +--+--+--+  +--+  +--+  +  +--+--+--+  +  +--+--+--+  + 
|           |  |     |     |           |  |           |  | 
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ 

The tiling group of L5:

The left-L5 is member of the right-L5 tiling group (Michael Reid)
This property gave hope to find an odd right-L5 rectangle.

  a       d 
 baaaa   ddd-- 
 bbbb    ddd- 
 cccc    ddd- 
    c       - 


 aab    --- 
 abbb   ---- 
 abbb   ---c 
 abbb   cccc   which gives the other orientation of L! 


The tiling group of P5:

Theorem P5(right) [Michael Reid]:

    x * y^5 * x^-1 * y^-5   is no relation of the group

    < x, y | x^3 * y^2 * x^-2 * y^-1 * x^-1 * y^-1,
             y^3 * x^-2 * y^-2 * x * y^-1 * x,
             x^-3 * y^-2 * x^2 * y * x * y,
             y^-3 * x^2 * y^2 * x^-1 * y * x^-1 >


Michael Reid was able to settle the one-sided P pentomino.  It does not tile
any odd rectangle, as expected.  Here's a homomorphism to S_80:
(The computations have been done by GAP.)

gap> x := ( 1, 4, 5, 8, 9, 2, 3, 6, 7,10)(11,24,35,62,71,12,23,36,61,72)
>         (13,26,41,56,63,14,25,42,55,64)(15,21,48,49,65,16,22,47,50,66)
>         (17,30,37,60,67,18,29,38,59,68)(19,27,46,53,69,20,28,45,54,70)
>         (31,44,51,73,78,32,43,52,74,77)(33,40,57,75,80,34,39,58,76,79);;
gap> y := ( 1,21,62,12,46)( 2,27,61,18,45,63,22,60,11,48)( 3,39,67,32,56)
>         ( 4,43,68,34,55,13,40,71,31,10)( 6,58,78,41,65,25,51,79, 8,70)
>         ( 5,52,77, 7,66)( 9,14,44,72,33)(15,38,75,30,53,20,35,74,23,50)
>         (16,36,76,24,54)(17,47,64,28,59)(19,37,73,29,49)(26,57,80,42,69);;
gap> x^3 * y^2 * x^-2 * y^-1 * x^-1 * y^-1;
()
gap> y^3 * x^-2 * y^-2 * x * y^-1 * x;
()
gap> x^-3 * y^-2 * x^2 * y * x * y;
()
gap> y^-3 * x^2 * y^2 * x^-1 * y * x^-1;
()
gap> x * y^5 * x^-1 * y^-5;
( 1,64)( 2,63)( 3,14)( 4,13)( 5,26)( 6,25)( 7,42)( 8,41)( 9,56)(10,55)(11,18)
(12,17)(15,20)(16,19)(21,28)(22,27)(23,30)(24,29)(31,34)(32,33)(35,38)(36,37)
(39,44)(40,43)(45,48)(46,47)(49,54)(50,53)(51,58)(52,57)(59,62)(60,61)(65,70)
(66,69)(67,72)(68,71)(73,76)(74,75)(77,80)(78,79)
gap> 


References:

 - Yuri Aksyonov
    Handed Y-pentomino problem is solved,
    email from 1999-03-06

 - David A. Klarner
    Solution to Problem E1543,
    AMM 70:7 (Sep. 1963) 760-61
    (prime rectangles for L4 and "one-sided" L4.)

 - Michael Reid
    There are odd "one-sided" L pentominoes prime rectangles,
    email from 1999-01-15

 - Michael Reid
    G hexomino [and handed P pentomino solved],
    email from 1999-04-14