Hello Igor!

What is the way you are writing the characters to the display?

I have been using lcds with HD44780 controller (I don't know your KS0066, but I expect, it is working nearly the same way). LCD display 4x20 needs 2 HD44780. The start addresses of each line looks as below:
line 1: 0 (up to 19)
line 2: 64 (..83)
line 3: 20 (..39)
line 4: 84 (..103)
If you are writing char's code to display (incremental mode) it displays line 1, then 3, 2, 4. If you want to write routine positioning the cursor, you have to add the offsets depending on your cursor position.

When I have started my first steps with lcd displays, I have made a cable between PC's centronics and lcd. I have wrote a simple program in Pascal or C and I was trying the lcd's function. It is very simply to do and Borland Pascal or C has really very good debugger, so you can look at all changing levels of data bus in step or real mode. You can write delays and test it. It is really very easy to do and to make some experiments. Sometimes it "clears" the problem with lcd.

Try to write one of character code (f.e. letter 'A') in a simple loop (send 80 chars to lcd) after initializing procedure. For example (for better understanding the idea):

CALL InitLCD
MOV R0,#80
LOOP:
MOV A,#'A'
CALL OutAccToLCD
DJNZ R0,LOOP

If your lcd displays all lines, it works good and you have to find error in your positioning routine.