SDHC microSD card and SPI initialization

I have a 32 GB Kingston SDHC microSD card which has to communicate to MSP430F2618 via SPI. I was not able to initialize it by using CMD55 + ACMD41 (with bit30 set to 1), as it is described in SD specification paper. What works is this sequence:

  CMD0, arg: 0         , CRC: 0x95 (response:0x01)
  CMD8, arg: 0x1AA     , CRC: 0x87 (response:0x01)
  CMD1, arg: 0x40000000, CRC: 0xFF (response:0x00)

I tried it also with a 8 GB SanDisk card and works for that too. So actually, I am using CMD1 here, not ACMD41, with HCS bit set to 1. However, it is not stable, sometimes it works, sometimes it does not. This may be about hardware.

Is it not strange that cards are responding to CMD1 with argument 0x40000000?

You shouldn't use CMD1 to initialize SDHC cards; as said in the SD Card specification:

In any of the cases CMD1 is not recommended because it may be difficult for the host to distinguish between MultiMediaCard and SD Memory Card.

Some controllers (newer and higher capacity cards mostly) will simply stay in IDLE if you issue CMD1 to them. You should first issue CMD8 0x1AA after the reset (CMD0), and then use CMD55 + ACMD41.

tl;dr to initialize the card you should:

CMD0 arg: 0x0, CRC: 0x95 (response: 0x01)
CMD8 arg: 0x000001AA, CRC: 0x87 (response: 0x01)
CMD55 arg: 0x0, CRC: any (CMD55 being the prefix to every ACMD)
ACMD41 , arg: 0x40000000, CRC: any
if response: 0x0, you're OK; if it's 0x1, goto 3.

Note that most cards require steps 3/4 to be repeated, usually once, i.e. the actual sequence is CMD0/CMD8/CMD55/ACMD41/CMD55/ACMD41 - to be sure, try it n times (select n within your reason), and assume fail if it doesn't.

Also, note that you should assert S̲S̲ (aka CS) low at least before and after CMD0 - and, in reality, probably always when sending any command to the SD card.