The specification doesn't seem to indicate how to pad messages to the requisite 512-bit boundary.
What is the padding scheme for the original Tiger hash?
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For drop-in compatibility, we adopt the outer structure of the MD4 family: the message is padded by a single '1' bit followed by a string of '0's and finally the message length as a 64-bit word. The result is divided into n 512-bit blocks.
I assume this 64 bit word is little endian, like the rest of Tiger and like the MD4 padding.
For reference the RFC1320 which describes MD4 says:
a sequence of bytes can be interpreted as a sequence of 32-bit words, where each consecutive group of four bytes is interpreted as a word with the low-order (least significant) byte given first.
Step 1. Append Padding Bits
The message is "padded" (extended) so that its length (in bits) is congruent to 448, modulo 512. That is, the message is extended so that it is just 64 bits shy of being a multiple of 512 bits long. Padding is always performed, even if the length of the message is already congruent to 448, modulo 512.
Padding is performed as follows: a single "1" bit is appended to the message, and then "0" bits are appended so that the length in bits of the padded message becomes congruent to 448, modulo 512. In all, at least one bit and at most 512 bits are appended.
Step 2. Append Length
A 64-bit representation of b (the length of the message before the padding bits were added) is appended to the result of the previous step. In the unlikely event that b is greater than 2^64, then only the low-order 64 bits of b are used. (These bits are appended as two 32-bit words and appended low-order word first in accordance with the previous conventions.)
At this point the resulting message (after padding with bits and with b) has a length that is an exact multiple of 512 bits. Equivalently, this message has a length that is an exact multiple of 16 (32-bit) words. Let M[0 ... N-1] denote the words of the resulting message, where N is a multiple of 16.