Q: How do I read or write registers?
This is from common.h:

#define REGISTER_WRITE( reg, data ) output8( reg, data )
#define REGISTER_READ( reg ) input8( reg )

(These input8 and output8 functions map (almost) 1:1 to a Z80 assembler instruction (IN and OUT?) for i/o.)

Q: What registers are there?
See the list below and also here.

Register	Hex value	Description
REG_MEMTYPE	0x00
REG_BANK1_LO	0x01	Determines the page mapped to $8000-$9FFF
REG_BANK1_HI	0x02	Determines the page mapped to $8000-$9FFF
REG_BANK2_LO	0x03	Determines the page mapped to $A000-$BFFF
REG_BANK2_HI	0x04	Determines the page mapped to $A000-$BFFF
REG_INT_REGDAT	0x05
REG_INT_REGRST	0x06	bitwise clear interrupt bits in REG_INT_REGDATA
REG_INT_MASK	0x07
REG_HALT	0x08
REG_FRC	0x09
REG_EXEC	0x0a
REG_EXT_CTRL	0x0b
REG_BLD_CTRL	0x0c
REG_KI_DATA	0x10
REG_KO_DATA	0x11
REG_P_DATA	0x12
REG_P_IOSET	0x13
REG_P_OUTSET	0x14
Beep sounds
REG_ALM	0x15	standard tone
REG_MELODY	0x16	custom tone
REG_MELFRQ_LO	0x17	freq low
REG_MELFRQ_HI	0x18	freq high
REG_MLDALM	0x19	switch standard/custom tone
REG_RMT_CTRL1	0x1a
REG_RMT_CTRL2	0x1b
REG_RMT_CTRL3	0x1c
REG_SIOCLK	0x1d
REG_RTC	0x1f
REG_SRLCD_DSP	0x20
REG_SRLCD_SEG	0x21
REG_SRLCD_LO	0x22
REG_SRLCD_HI	0x23
REG_SRLCD_COM	0x24
REG_RMLCD_SEG1	0x25
REG_RMLCD_SEG2	0x27
REG_RMLCD_SEG3	0x29
REG_RMLCD_COM	0x2b
REG_DRV_CTRL	0x2d
RTC = Real Time Clock
REG_RTC_1SEC	0x30
REG_RTC_10SEC	0x31
REG_RTC_1MIN	0x32
REG_RTC_10MIN	0x33
REG_RTC_1HR	0x34
REG_RTC_10HR	0x35
REG_RTC_DAY	0x36
REG_RTC_1DAY	0x37
REG_RTC_10DAY	0x38
REG_RTC_1MON	0x39
REG_RTC_10MON	0x3a
REG_RTC_1YR	0x3b
REG_RTC_10YR	0x3c
REG_RTC_PAGE	0x3d
REG_RTC_TEST	0x3e
REG_RTC_RESET	0x3f
REG_RTC_24	0x3a
REG_RTC_URU	0x3b
SIO=Serial Input/Output
REG_SIO_RDB	0x40	See below
REG_SIO_TDB	0x40
REG_SIO_DLL	0x40	DLL, Division Latch LSB: port 0x40 = 0x07 //
REG_SIO_IER	0x41	IER, Interrupt Enable Register: port 0x41 = 0 // this disables interrupts.
REG_SIO_DLM	0x41	DLM, Division Latch MSB: port 0x41 = 0x00 // 9600 baud or near enough
REG_SIO_IIR	0x42
REG_SIO_LCR	0x43	LCR, Line Control Register: port 0x43 = 0x80 // access division latches =0x03 // 8 bit, No parity, No stop bit
REG_SIO_MCR	0x44	MCR, Modem Control Register: port 0x46 = 0x00
REG_SIO_LSR	0x45
REG_SIO_MSR	0x46
REG_SIO_SCR	0x47
LCDC I/O
REG_LCEN	0x50
REG_LCMD	0x51
Key Scan */
REG_SCNEN	0x60	SCaN ENable?
REG_ATPC	0x68
REG_COLD	0x69	Touch screen data - see below
REG_COLU	0x6A	Touch screen data - see below
REG_ROWD	0x6B	Touch screen data - see below
REG_ROWU	0x6C	Touch screen data - see below

0x01-0x04 : Memory paging
Memory paging on REX is done by IO registers 1-4 in this order:

IO port 1 = bank 1 LOW (bit 14 to 21)
IO port 2 = bank 1 HiGH (bit 22 to 25)
IO port 3 = bank 2 LOW (bit 14 to 21)
IO port 4 = bank 2 HIGH (bit 22 to 26)

Bank1 starts at $8000 and bank 2 at $a000. Both are 8192 bytes long.
If you write the following values:
port 0x03 = 0x00
port 0x04 = 0x10
you see the normal 8k RAM at address 0xA000 and
port 0x03 = 0x01
port 0x04 = 0x10
you see a second 8k RAM at addres 0xA000 which can apparently be used safely.

0x15-0x19 : Beep sounds
Used as follows:
REGISTER_WRITE(REG_MELFRQ_HI,frqhi);
REGISTER_WRITE(REG_MELFRQ_LO,frqlo);

The freq parameters are not actually frequencies, but the inverse (wavelength?). The higher the frq the lower the tone the high-frequency is best set to 0 to produce audible beeps. with frq_hi>0 a low tone is produced.

REGISTER_WRITE(REG_MELODY,2); // set command??
REGISTER_WRITE(REG_MELODY,1); // play command

REGISTER_WRITE(REG_MLDALM,1); //choose melody instead of alarmsound
REGISTER_WRITE(REG_MLDALM,0); // default to alarm sound

REGISTER_WRITE(REG_ALM,0); ?

0x22-0x23 : LCD display
port 0x22 = 0x00
port 0x23 = 0x10
displays the first 8k RAM on the LCD,
port 0x22 = 0x01
port 0x23 = 0x10
displays the second 8k RAM on the LCD.
This way you can show any 8k on the LCD, even flash memory.

0x40-0x47 : Serial Input Output
The rex appears to have a basic 16450 UART (or similar) which can be controlled by registers 0x40-0x47
z80.h with definitions of all IO Docs about using serial ports should not be a problem (eg., http://www.lammertbies.nl/comm/info/RS-232_uart.html)
It is indeed somewhat like the stuff you see in a PC so you can use it much the same. The clock that is used to drive it is different though which leads to different values for the division latches. (probably they saved a crystal by reusing the signal used for the processor or RTC)

Now you can send data by sending out'ing it to port 0x40 and receive it by reading from port 0x40. Use the LSR (line status register) to see if the byte has been sent and a new one can be sent and to see if a byte can be read. LSR bit 0 : data available
LSR bit 1 : overrun error (previous data was overwritten)
LSR bit 2 : parity error
LSR bit 3 : framing error (usually baudrate difference between sender & receiver)
LSR bit 5 : byte was sent (can write next byte).
Funny thing is that the Rex is capable of higher speeds than 9600 baud, but because the Rex uses a divider of 7, and the PC uses a divider of 12, there are some troubles getting it to work.

Known working was at 57600 baud which is the same a RexTools appears to use. Apparently the Rex uses by default a divisor of 1, giving about 67200 baud (this is not an exact science !!) together with a setting of 2 stopbits & low parity. If you use 57600 baud on the PC and 1 or 2 topbits with low parity, you can actually communicate with the occasional parity error (!!!) but that may be why a checksum is needed?

0x69-0x6C : Read the touch screen
This might be useful for games or applications (drawing?) that require animation or speed in general, since event-processing (DsEventMessageGet(&msg);) takes some time. They contain the (uncalibrated?) touchscreen position, or 0 if the screen isn't touched.

The 16-bit value ranges Rex are approximately (decimal):
Y: 80 (bottom), 862 (top)
X: 62 (right) , 940 (left)
Note that the values range in the opposite direction to the screen coordinates.