Given that half a child's genes are from his/her mother and half from
his/her father, is it theoretically possible for offspring of the same
parents to have completely different genes?  What i mean is, say the
mother's genes are A and B, and the father's C and D, can one child be
A+C and another B+D?

Well as per my understanding goes the siblings and parents genetic
structures have similarity.The siblings inherit the genetic structurs
of their parents.

Let me explain it in this way :

Suppose mothers genetic struture is AB and fathers is CD

Siblings inherit the structure AC or AD or BC or BD.

That means that the siblings have a different genetic structure
between themselves but linked to the same root.That is if the two
sibling's genetics are analysed and if one of their parents genetic
structure is known it can be found out if they are really from same
origin i.e. both of them have atleast one common parent.The trace can
go deep down to analysis of DNA and it will be found that at least one
DNA of the siblings match if they are really siblings.

Please also note that the new structures in sibling are formed by a
process called mutation ( genetic mutation).

Useful link :

http://www.synapses.co.uk/genetics/repflow.html

http://ghr.nlm.nih.gov/info=mutations_and_disorders/show/mutations_cause_disease

In theory it is possible that two siblings have completely different
sets of chromosomes. However, given that the chance for any
combination of a single chromosome pair is 1:4, and that humans have
23 chromosomes, the probabilities are very low.

In addition applies that not all genetic information is transfered by
nuclear chromosomes, and that mutation does occur.

The quick answer is it isn't going to happen, though it is not
theoretically impossible. Each parent has 23 pairs of chromosomes. The
pairs are similar, but not identical, and in the formation of egg or
sperm one of each pair is picked at random. The odds of two eggs
getting different copies of a given chromosome is 1/2: so the odds
that they get all 23 different is 1/2 x 1/2 x 1/2..........23 times,
or about 1 in 8,000,000. The odds that two sperm also contain all
different picks is likewise 1, in 8,000,000. The chances of those two
meeting is 1 in 64,000,000,000,000. In other words, if every person on
earth had come from the same two parents it is still virtually
impossible that two people would have receievd all different (or all
the same)genes.
In reality the odds are MUCH worse. In forming eggs and sperm the
pairs of chromosomes randomly exchange some of their genes, which has
the effect of making the problem act as if there were over a hundred
pairs, rather than 23, so the numbers go up to only one in 10^30 egg
or sperm being all the same, or different and only one in 10^60 people
having all the same or different genes.

NEVER ANY CHOICE AT ALL.

I feel the way you framed the question itself is wrong. 

Lets see it here?
Parents father and mother have most of the genes same but differ only
in few genes compared to the whole number of genes a human being is
composed of. That is what differentiates a female from male.
So if u take half of the genes from mother and half from father and
form a sibling, she/he would never have all different genes .
Because the   more than 50% of genes in male and female are similar so
u cannot expect to construct an offspring by taking genes from mother
which are not present in father.

A LAYMANS EXAMPLE
Suppose you have two buckets marked as A and other as B.
The bucket A has 60 red balls and 40 balls different coloured
The bucket B has 60 red balls and 40 balls different coloured.
So if I ask u to take 100 balls, 50 from each.
Can u expect to have all the 100 balls differently?
NEVER
OK
I think this would solve your question
I did not go into details, as I don?t know your background.
But it?s a good question if you?re not from biology.
Comments are most welcome
A NOTE:
Actually the difference between male and human genome is 3% only that
is 97% is same. I took its as 60% just to make it simple.